How does "The Yellow Wallpaper"'s syntax and diction depict the narrator's deteriorating sanity?

Charlotte Perkins Gilman uses both syntax and diction masterfully to show the protagonist's deteriorating mental state in the short story "The Yellow Wallpaper."

Syntax refers to the arrangement of words and phrases in order to produce well-constructed sentences. 

"He has no patience with faith, an intense horror of superstition, and he scoffs openly at any talk of things not to be felt and seen and put down in figures. John is a physician, and perhaps—(I would not say it to a living soul, of course, but this is dead paper and a great relief to my mind—) perhaps it is one reason I do not get well faster."

Notice the sophisticated sentence structure or syntax. The narrator is speaking about her husband and gives a list of three qualities he has, using phrases and vocabulary that depict a fair amount of intelligence. In the second sentence, she interrupts her thoughts to intimate that she would not share these ideas with anyone. Her sentences in the beginning of the story are complex, and they give the reader the sense that she is educated and intelligent. Readers would have no concern about her mental state at this point in the story other than the fact that she divulges that she is sick with a "nervous condition." 

If a physician of high standing, and one's own husband, assures friends and relatives that there is really nothing the matter with one but temporary nervous depression—a slight hysterical tendency—what is one to do?

This quote is also toward the beginning of the story. The narrator is still using complex sentences with appositives and sophisticated syntax. This makes her seem intelligent and fully in control of her mental faculties.

As the story goes on, we begin to see a change. The narrator is writing more simple sentences. The content of the sentences sounds more like the writing of a willful child than an educated adult woman: 

I don't know why I should write this.

I don't want to.

I don't feel able.

Later, she becomes obsessed with the patterns in the wallpaper and thinks she sees a woman creeping around behind the pattern. It isn't just her sense of sight that is given over to the obsession of the creeping woman; the paper also has a smell that becomes a pervasive element in her psyche. In the quote below, she is describing the smell of the wallpaper. Notice the short, simple sentences. The vocabulary is simple. She has personified the smell. This all points to a deteriorating mental condition.

"It creeps all over the house. I find it hovering in the dining-room, skulking in the parlor, hiding in the hall, lying in wait for me on the stairs. It gets into my hair."

Toward the end of the story, her vocabulary (diction) is very diminished. The author's word choice indicates a woman who is regressing. She is creeping, which means to move very slowly and close to the ground. Her words and actions cause her husband to faint because he realizes that she has lost her mind. The syntax is also very simple and very different from the syntax used in the beginning of the story. 

I kept on creeping just the same, but I looked at him over my shoulder. "I've got out at last," said I, "in spite of you and Jane. And I've pulled off most of the paper, so you can't put me back."

In "The Yellow Wallpaper," how does Gilman's characterization show the woman's mental decline?

Let's begin by reviewing the definition of "characterization." Basically, characterization in literature reveals the author's creation and representation of a literary character with his or her unique physical and psychological traits. Characterization involves four main elements: description, a presentation of the character's thoughts and feelings, dialogue, and the character's reactions to other people. Charlotte Perkins Gilman uses all of these in "The Yellow Wallpaper" to show the narrator's descent into madness.

While we never receive an explicit description of the narrator's appearance, we quickly understand that she is a young married woman with a baby who is suffering from a nervous condition. Her husband has brought her to a country house for relaxation, and as her physician, he tries to control her activities completely.

Gilman is especially descriptive with regard to the narrator's behavior, which becomes more and more bizarre as the story progresses. At first, she is merely fixated on the wallpaper in her bedroom, but later, she begins to see a woman in it. She even gets up at night to see if it has moved. Eventually, she tears the paper off the wall, trying to free the woman inside, and by the end of the story, she is creeping around the room, positive that she herself is the woman in the wallpaper and that she will not allow anyone to put her back inside it.

Gilman also makes effective use of the narrator's thoughts and feelings, which are related in the first person. Through these especially, we can watch her decline into madness. At the beginning of the story, the narrator is more frustrated and nervous than anything else. She experiences anxiety, and while she tries to convince herself that her husband loves her and is doing right by her, she chafes under his rules and restrictions. He will not allow her to have visitors or to write, and this actually increases her nervousness rather than helping it.

Bored, the narrator turns her attention to the wallpaper and becomes more and more obsessed with it and with the woman she sees within it. The woman, of course, is symbolic for the narrator herself, and she comes to identify with the woman. She eventually locks herself in the bedroom so she can creep freely.

There is not a lot in the way of dialogue in "The Yellow Wallpaper," but what there is certainly gives us a glimpse into the narrator's psyche and her husband's plan of treatment. Her husband is patronizing when he speaks to her, treating her much more like a child than a grown woman. When she asks him to move their sleeping quarters downstairs, he refuses. He calls her "little girl," orders her around, and says they can't possibly go home yet. Everything his wife says, he laughs at and brushes aside as silly. It would be enough to drive anyone crazy.

The dialogue between husband and wife at the end of the story is especially revealing, for when her husband finally enters the bedroom, the narrator yells out, "I've got out at last in spite of you and Jane! And I've pulled off most of the paper, so you can't put me back!" In the narrator's mind, she is the woman behind the wallpaper.

Finally, the narrator's interactions with other characters also shows her increasing mental difficulties. She becomes paranoid, believing they are trying to thwart her activities concerning the wallpaper. She doesn't want them to even look at it or to take her away from her meditations upon it. Finally, she locks everyone out of the bedroom so she can creep as she pleases, and when her husband faints, she merely creeps right over him.

In "The Yellow Wallpaper", how does the narrator's illness progress from beginning to end?

At the beginning, the story phrases it as a "temporary nervous depression".  So, they isolate her in the country away from socializing, don't let her write, and keep in a bedroom that she absolutely hates.  At first, she is fine; she is optimistic.  2 weeks later, she is getting worse.  She mentions John being gone a lot, and that "these nervous troubles are dreadfully depressing.  John does not know how much I really suffer."  She can barely do the littlest things, and "cannot be with" her baby.  A while later after the 4th of July, she has digressed to, "I don't feel as if it was worth while to turn my hand over for anything, and I'm getting dreadfully fretful and querulous.  I cry at nothing, and cry most of the time."  It is soon after this that she describes the first of her "hallucination" of the woman behind the wall, so she is rapidly deteriorating.  In the next entry she is obsessed with the paper, "a little afraid of John" and acting paranoid towards his sister.  Finally at the end, she is completely paranoid; she's locked herself in the room, and is crawling around the edges of the walls, just like the woman in the wallpaper used to.

She starts off as being slightly nervous or depressed, and digresses to complete paranoia, hallucinatory imaginings, hostile behavior, and removal from reality.

How does the setting contribute to the woman's descent into insanity in The Yellow Wallpaper?

The setting of "The Yellow Wallpaper" both reflects the narrator's mental isolation and plunges her further into her own world of mental struggles.

Consider how the house's physical isolation is described near the beginning of the story:

The most beautiful place! It is quite alone, standing well back from the road, quite three miles from the village. It makes me think of English places that you read about, for there are hedges and walls and gates that lock, and lots of separate little houses for the gardeners and people.

The house is beautiful, but it restricts the narrator from any contact with the world around her. Set in the late 1800s, the distance from the village would be significant enough to make social gatherings difficult without proper planning, and her health isn't stable enough for those efforts. There are both walls and gates that lock, and it seems almost like the description of a jail. Also noteworthy is the way workers are kept far from her. She is subjected to near total isolation from almost everyone except those whom her husband grants permission to visit. She deeply desires further human contact, but her husband won't allow it:

I tried to have a real earnest reasonable talk with him the other day, and tell him how I wish he would let me go and make a visit to Cousin Henry and Julia. But he said I wasn't able to go, nor able to stand it after I got there and I did not make out a very good case for myself.

The house becomes her prison more and more as the story progresses, noted by her projection of a woman imprisoned within the wallpaper in her room.

The narrator is also confined to the room which was formerly a nursery. This is both symbolic of how her symptoms are misunderstood and even trivialized by her husband and a further source of angst considering that she suffers from postpartum depression. When speaking of her baby, the narrator notes that "[she] cannot be with him, it makes [her] so nervous." It seems that a woman who could not bear to be with her baby would be allowed to rest in a room other than a former nursery.

The room itself stands in sharp contrast to the world she is allowed to view from her windows. Her room is dreary:

Then the floor is scratched and gouged and splintered, the plaster itself is dug out here and there, and this great heavy bed which is all we found in the room, looks as if it had been through the wars.

Abuse and neglect stare at the narrator from every corner of the room, yet she can see a different world outside because the room has many windows, allowing her to view in all directions the nature that is just beyond her reach:

Out of one window I can see the garden, those mysterious deep-shaded arbors, the riotous old-fashioned flowers,
and bushes and gnarly trees.

Out of another I get a lovely view of the bay and a little private wharf belonging to the estate. There is a beautiful shaded lane that runs down there from the house. I always fancy I see people walking in these numerous paths and arbors, but John has cautioned me not to give way to fancy in the least.

The narrator clearly connects with the beauty of nature, but she cannot allow her soul to enjoy it directly. Instead, her husband keeps her confined to her dilapidated nursery—yet constantly reminded of what she could be enjoying.

The physical isolation, the choice of a nursery for a woman suffering from postpartum depression, and the separation from nature all contribute to the narrator's mental decline.

Does the woman in "The Yellow Wallpaper" lose control of her mind?

One element to keep in mind about "The Yellow Wallpaper" is that it is written in the first person perspective. This entire story is told as a series of journal entries, written by the story's main character. Therefore, I'd suggest that the story's style of writing, and its method of presentation, represent a critical resource for answering the kind of question you have posed.

With this in mind, I think it's especially important to note that, as far as "The Yellow Wallpaper" is concerned, at a certain point in the story, the tone of her entries actually shifts, and her writing becomes far more erratic and manic in presentation. If you want to find evidence that the main character is losing her mind, I would suggest that one place you could look is in the story's use of language and its style of presentation, which serves to reflect its narrator's own mental state and the mental deterioration she experiences. This is one of the central conceits concerning how first person point of view functions.

Does the woman in "The Yellow Wallpaper" lose control of her mind?

I have had to edit your question to only ask one question. Please remember that enotes does not permit you to ask multiple questions. However, thinking about the theme of insanity in this excellent short story, you will want to analyse how the woman comes to identify her own situation with the wallpaper, in the end creating an alter-ego that is trapped behind the "bars" of the wallpaper that she "releases" at the end of the tale. The woman is clearly projecting her own sense of being confined, emotionally, intellectually and otherwise, into the wallpaper, and this can be traced with the growing relationship that the narrator experiences with the wallpaper. Perhaps the most tragic moment of this short story is at the end, when this identification becomes complete, and the narrator sees herself as that woman trapped, and says to her husband:

"I've got out at last," said I, "in spite of you and Jane! And I've pulled off most of the paper, so you can't put me back!"

However, it is tragically ironic that although she declares her freedom stridently, her continued creeping round the edge of the room undercuts her protestations of freedom, making it clear that she has descended into new depths insanity. Whilst she may have "freed" the woman from the second layer of the wallpaper, this "prison" has now become real for her, as she assumes the persona of the trapped room and is left to wander around the edge of it.

How does the narrator in "The Yellow Wallpaper" lose touch with reality and become insane?

I would begin with an introductory paragraph explaining three key indicators of the narrator’s insanity: the change in style of the narrative, the change in the narrator’s attitude and the transformation of the wallpaper itself.

 In the second paragraph you can comment on how the narrative changes from a detailed account of the house itself, the gardens and life beyond the room to a fragmented account of the narrator's internal hallucinations. From –

The most beautiful place! It is quite alone…

 To-

 …outside you have to creep on the ground, and everything is green instead of yellow.

The third paragraph could illustrate how the narrator goes from begrudgingly obeying her husband to defying him by the end. As she concludes-

 I had to creep over him every time!

Paragraph four could show the wallpaper changing from the narrator’s focus, to fascination, to prison-

 I've pulled off most of the paper, so you can't put me back!

 The fifth paragraph concludes the essay explaining that these three elements demonstrate the narrator’s decline into insanity.

How does the woman in "The Yellow Wallpaper" gradually become mad and insane?

The insanity that comes on the woman in this excellent classic is definitely something that grows gradually. It starts of with the confinement of the narrator to her room for "rest," which, according to John, the narrator's husband, is all that she needs. This is typical of the views surrounding depression of the time, and the narrator is confined to her bed and not allowed to see her baby. However, the first section ends with the narrator expressing her dislike of the wallpaper:

It is dull enough to confuse the eye in following, pronounced enough to constantly irritate and provoke study, and when you follow the lame uncertain curves for a little distance they suddenly commit suicide--plunge off at outrageous angles, destroy themselves in unheard-of contradictions.

Note how this impression of the wallpaper develops gradually. At the end of the second section, the narrator discerns a "strange, provoking, formless sort of figure" that exists in the second layer of design. Gradually, as the narrator's obsession with the wallpaper intensifies, she says:

There are things in that paper that nobody knows about but me, or ever will.

She has discerned that the figure in the wallpaper that she sees is "like a woman stooping down and creeping about behind the pattern." Then the narrator becomes paranoid, thinking that John and Jennie want to discern the secret of the yellow wallpaper before she does. However, the wallpaper now seems to be a cause for health in the narrator, rather than something that overtly makes her worse. However, the narrator detects the "smell of the wallpaper" permeating the entire house, which is a "yellow smell." The narrator finally can discern that the exterior pattern on the wallpaper is moving because the woman trapped behind it shakes it:

And she is all the time trying to climb through. But nobody could climb through that pattern--it strangles so; I think that is why it has so many heads.

Finally, the narrator decides to free the woman trapped behind the wallpaper. One day, she locks her room and throws the key out of the window. Although she pulls a lot of the paper off the wall, and the story ends with the disturbing image of the narrator creeping around the edge of the room by the wallpaper, having to step over the fainted body of her husband at each circuit.

Note how the increasing identification with the wallpaper creates a kind of Gothic "double" whereby the narrator projects her own situation into the life of the woman who is trapped behind the wallpaper.

What are examples of madness in "The Yellow Wallpaper"?

The most conventional approach to this answer lies in finding the examples of madness that the woman experiences in her time in the room.  One example of this would be how what the woman feels is needed for her is constantly discarded by the men around her.  The madness in this results in the collision between her wants and the decisions of the husband and the doctors about what she needs. Madness is evident in the obsession about the wallpaper.  In representing the only source of focus for the woman, one can see elements of madness.   The "eyes" from the wallpaper that she perceives as staring back at her is reflective of such a condition.  Additionally, the figure that she sees "skulking behind the pattern of the wallpaper" is another example of the madness that is being fostered and facilitated within her.  There is a type of madness in which the narrator thinks that the wallpaper is a reflection of her own identity, in the way she sees the old woman within its patterns and in how she traces one line of it until its conclusion.  The focus on the wallpaper is the result of the only thing towards which she can channel her energies. Madness develops from such a condition in which she "becomes" the woman inside the wallpaper, crawling around the room to be free.

Another form of madness might be embodied by the emphasis in "rest therapy."  The doctors and husband who seem to believe in the certainty of this therapy and the negation of other approaches to help the woman represent the same singular madness that the woman suffers.  There is madness in the resistance to validate the woman's voice and even try to integrate some of the things she wants to do such as walks outside, reading, or writing into the notion of "rest therapy."  Madness can be seen in the singular and monolithic focus where there is little perspective evident.