her father quickly.

This seems to show that Emily has had some sort of mental breakdown. After Emily's death, a man's body is found (presumably Homer Barron's), and the presence of a grey hair on the pillow suggests Emily had been laying next to the decaying body. This certainly can be used as evidence for insanity.

On the other hand, one could argue against insanity by referring to Emily's encounter with the druggist. Emily purchases a strong poison, specifically arsenic. She only says it is "for rats." These seems to suggest Emily's murder of Homer is calculated, and she is smart enough to claim it will be used to get rid of vermin.

If Emily is not insane, how could we explain her eccentric behavior? Well, for starters, Faulkner describes Emily's heritage and how she is a product of the Old South. Insane or not, her upbringing and treatment from her father has certainly had an impact on her life.

We should note that Emily has encountered illness:

She was sick for a long time. When we saw her again, her hair was cut short, making her look like a girl, with a vague resemblance to those angels in colored church windows—sort of tragic and serene.

We do not know what this sickness was, so perhaps it affected her mind and/or mental clarity.

Emily exhibits memory issues in her interactions. Emily denies her father's death:

She told them that her father was not dead. She did that for three days, with the ministers calling on her, and the doctors, trying to persuade her to let them dispose of the body.

She displays similar behavior when the aldermen come to collect her taxes:

“See Colonel Sartoris.” (Colonel Sartoris had been dead almost ten years.) “I have no taxes in Jefferson."

Could these memory issues reveal an illness such as Alzheimer's disease, that is causing Emily to behave this way? Are these examples of a calculated manipulator, saying what she knows will get people to leave her alone? Or is she truly insane?

I would think that this would depend on the definition of "insane" that you are using when you answer this question.  I think that you can argue it either way depending on how you define the term.

On the one hand she clearly does not behave in any sort of a normal way.  She denies that her father is dead for a few days before she finally allows him to be taken away.  Much worse, she seems to have been sleeping in the same bed as Homer Barron's remains for decades.  Surely someone who kills a person and then sleeps in the bed with their corpse is not completely sane.

But, at the same time, she is clearly able to function and to know what society will think of her (to some extent).  She makes sure that people do not know what she has done (by, for example, not telling the druggist why she wants arsenic).  She teaches art for a while.  She rebuffs the city leaders when they try to get her to pay taxes.  She is clearly not insane in the sense of being unable to function in her society.

William Faulkner's Southern Gothic masterpiece, "A Rose for Emily," is disordered in time, perhaps as a reflection of the disorder in particular characters' minds.

The titular character, Miss Emily Grierson, suffers from a disorder of time. That is, she suffers from a mad hold on the past. For example, she is firmly convinced that Colonel Sartoris, who was once the mayor of the town, has provided for "a dispensation dating from the death of her father on into perpetuity." In other words, Miss Emily believes that she never has to pay taxes because Colonel Sartoris made arrangements that the Griersons were exempt from paying taxes. But when the "next generation" in town becomes aldermen and mayor, they know nothing of such an arrangement. Nevertheless, when a deputation visits her, Miss Emily insists,

"I have no taxes in Jefferson.... Perhaps one of you can gain access to the city records and satisfy yourselves."
"But we have. We are the city authorities, Miss Emily. Didn't you get a notice from the sheriff signed by him?"
"See Colonel Sartoris. I have no taxes in Jefferson."
"But, Miss Emily—"
"See Colonel Sartoris."

Colonel Sartoris, however, has been dead about ten years.

Of course, the strongest example of madness in Emily Grierson is revealed at the end of Faulkner's story. After Miss Emily's death in a room downstairs, the ladies of town arrive and are given entrance to her home by the old servant. Later, after Emily is buried, the townsfolk return and enter a room upstairs. There they find evidence of Emily's madness when they discover the rotted body of Homer Barron lying on a bed and covered with dust. To add to their horror, the townsfolk discover the indention of a head on the other pillow, and on this pillow is a long strand of "iron-gray hair" that apparently belonged to Miss Emily.

William Faulkner's "A Rose for Miss Emily" depicts madness in many forms and through many characters.

Miss Emily's father demonstrates a kind of madness when he refuses all of his daughter's suitors because they were not good enough; clearly he was stuck in some notion of the past and could not be realistic about the present.

Homer Barron must have been a bit mad to keep coming back to Miss Emily; he had many indications that he should have left and did not do so.

The town authorities clearly display a kind of madness when they are so thoroughly intimidated by Miss Emily that four grown men slink around her house after midnight, sniffing the ground and spreading lime, hoping Miss Emily will not catch them. Or when they allow her not to pay taxes like everyone else just because she looks at them imperiously and says she will not do so.

Obviously there is plenty of madness to go around in this story, but of course Miss Emily displays the most madness. She keeps her father's dead body for three days before the authorities can finally take it from her. After she and Homer Barron scandalously become lovers, the ladies in town force the Baptist minister to go and talk to her; he never reveals what happened during that interview, but it had to have been outrageously mad since he refused to enter Miss Emily's house again--ever. Her three cousins have the same kind of reaction when they come to chide her for her scandalous behavior. They do not stay long.

Miss Emily certainly displays a kind of madness when she boldly purchases poison and lies about why she needs it; and when she keeps her dead lover's body for years and then sleeps next to him (at least sometimes), she must also have been mad.

Some of what what I am calling madness may just be the quirkiness of an aging Southern belle (a kind of relic) who has been robbed of an opportunity for happiness and therefore prefers her own company to others'; however, Miss Emily also certainly demonstrates madness in some of its most gruesome forms. While it is easy to point to the grotesque and macabre things Miss Emily does, she does many other things worthy of the term "madness." 

This question has been previously asked and answered. Please see the links below, and thank you for using eNotes.

I loved this question! If you are writing an essay on this matter, however, be aware of the use of the word "crazy" if you are going to include it in your narrative. In this day and age everything has a politically-correct terminology...DSM IV can definitely help you find her an accurate diagnosis ;) .

But: To answer your question directly- In the years I've taught this story, I have not been able to come to terms with Emily being "crazy."

She is the Faulkner ideal persona of the Old South, itself: The hard-headed, norm-oriented, value-based, old school-ridden individual with a natural inability to accept what has happened: Change.

Surely, the consequences of her thoughts are irrational, criminal, and lead us all to do deem her as "crazy". Yet, if you look at the rationale behind her actions, her "craziness" can actually be labeled under "post traumatic stress" if you want to be real anal about it; the trauma of losing her father, and seeing the world (as she knows it) fade away steadily and leaving her with nothing, and nobody, as  support systems.

Concisely- Emily is a traumatized human being who, in her natural state, was borne and raised to be head-strong, value-based, and co-dependent. There is nothing terminally "crazy" about a human being who acts (even if irrationally) in defense of what he or she was made to believe to be reality.

Of course, normal behavior does not include poisoning one's lover and then sleeping in the bed with his corpse.  Thus, it is apparent that "Rose for Emily" is a gothic tale. And, while she is a victimizer in her dealings with Homer Barron, Emily is clearly a victim of the repression of the old patriarchal environment in which her father dominates  her; so, there is more to her character than merely being "crazy."  For, such a repressive life brings about aberrant behavior as one may seek to maintain some personal power against of his/her situation. 

It is this struggle to maintain personal power is exactly what motivates Emily.  Having lost her family, friends, and way of life of the Old South, Emily cannot accept any more loss.  Consequently, she retains her final illusion by not releasing Homer to the modern world in which she is an anachronism.

That Emily has detained her old world in her ancient house is evidenced in final passages of the story:

She died in one of the downstairs rooms, in a heavy walnut bed with a curtain, her gray head propped on a pillow yellow and moldy with age and lack of sunlight....[in another room] The man himself lay in the bed...The body had apparently once lain in the attitude of an embrace, but now the long sleep that outlasts love...had cuckolded him.