What are the conflicts in chapters 13-15 of The Catcher in the Rye?

One of the main conflicts present in the chapter is the constant search on Holden's part for companionship or some kind of way to alleviate his loneliness.  He tries to befriend the cab driver and basically fails, but he also isn't willing to take advantage of the offer of a seat next to D.B.'s former girlfriend, as of course the offer is "phony."

This of course leads to his encounter with the elevator man and his decision to order a prostitute.  This of course leads to him also refusing her services and then refusing to pay her and further issues there.  This is one more example of Holden being unable to act on his professed beliefs.  He can't just tell the girl that he doesn't think they should have sex because they aren't in a relationship so he uses the excuse of a recent operation.

What are the conflicts in chapters 18-22 of The Catcher in the Rye?

In chapters 18-22 of The Catcher in the Rye, Holden's plummets toward self-destruction, but he wisely chooses not to suicide like James Castle.  In these pivotal chapters, the book seems destined toward tragedy, but Holden's conversations with Antolini and his sister move the resolution (though open-ended) back toward comedy.

Holden is a sadist.  He baits Carl Luce to beat him up just like he got Stradlater and Maurice to punch him.  Holden wants someone to punish him for some unresolved guilt (possible survivor's guilt over his brother's death).  Luckily, Luce leaves.

Then, Holden goes to Mr. Antolini's, who, as a counter-culture spokesman, seems set up to be the Deus ex Machina (savior) of the novel.  He gives Holden great advice, and his voice seems to be that of the author's: he says Holden is “in for a terrible fall.”

The mark of the immature man is that he wants to die nobly for a cause, while the mark of the mature man is that he wants to live humbly for one.

Remember, Antolini was the one who found and carried James Castle's body to protect him from the rubber-necks.  This is want Holden wants: to commit suicide, and he's going to Antolini's to see how it will play out.

You see, James Castle (J. C. --“Jesus Christ”) is Holden’s martyred saint.  Castle fell to his death rather than take back any of the words he said about others (He called them "conceited").  Similarly, Holden doesn't want to take back calling people "phonies."  He wants to die young, a misunderstood romantic hero like Mercutio (from Romeo and Juliet), another non-phony.

After this, Salinger discredits Antolini by having him make a pass at the boy.  So, the episode is ironic, weird, and ultimately anti-climatic.  Holden continues his episodic journey toward being a catcher in the rye.

Holden knows his role as a catcher is doomed to fail.  He can't even protect Phoebe from the evils of the world (the f-bomb).  Holden finally realizes that he must enter the adult world.  So, he goes home to get mental health treatment.

What are the conflicts in the first four chapters of The Catcher in the Rye?

There are lots of conflicts in these chapters -- Holden against just about everyone he ever meets, essentially.  But it all boils down to two conflicts that continue through the whole book.  The two conflicts are Holden vs. society as a whole and Holden against himself.

We can see Holden in conflict with what society expects from him most clearly in Chapters 1 and 2.  We see this from the fact that he has been thrown out of Pencey and we see it in the way that he interacts with Mr. Spencer.  We also see him in conflict with Ackley and Stradlater.

A lot of this is because he is not sure what or who he wants to be.  We can see this partly in how he has the red hat.  He pretends to not care what society thinks of him, but there are times when he won't wear the hat so as not to stick out too much.

Are conflicts resolved at the end of The Catcher in the Rye?

Catcher in the Rye's is open-ended.  Salinger gives provides an authorial voice at the end (Mr. Antollini), but he doesn't save Holden.  So, in the end, the reader can make either case: Holden does mature and Holden does not mature.  It's an anti-climatic ending full of regret as Holden says that he misses everyone.  By the end, the reader is back where s/he started.

Here's what Enotes says about the ending:

Chapter 26 functions as an epilogue to the book. It is outside the story. Epilogues usually help explain what went before, but Salinger leaves this analysis to the reader. Whether Holden ended up as a Wall Street attorney, working at his father’s firm, or is, in fact, still frequenting Central Park, cursing the absurdities of life, nobody knows. Maybe Salinger knows, but he has said nothing—in over 40 years.

Here are my lecture notes (with quotes) regarding the final chapters:

I. No Deux ex Machina

A. literally “God out of the machine”

B. literary definition: “a plot device or character that rescues a hopeless situation”

C. Mr. Antollini: voice of Salinger; Mr. counter-culture professional

D. has a chance to rescue Holden

E. His advice:

1. Holden is “in for a terrible fall”

2. "The mark of the immature man is that he wants to die nobly for a cause, while the mark of the mature man is that he wants to live humbly for one."

F. Salinger discredits Antolini in the end by casting him as possible gay pedophile (no rescuing; no morals)

G. Anticlimactic ending: does Holden ever grow up? "don't ever tell anybody anything. If you do, you start missing everybody" (shows regret; a downer)

Is the conflict in The Catcher in the Rye satisfactorily resolved at the end?

This is an opinion question.  In my opinion, no, the ending of the story does not settle the conflict in a satisfactory way.  Holden is one of the most frustrating characters that I have ever had the privilege to read about.  He complains about everything.  He has his bright spots, but I found him exhausting.  Everything bothers him or is a pet peeve of his, and he fails to notice that the common denominator is him.  Maybe he's the problem.  Of course Holden would tell me to get lost since I'm such a phony.  He can't stand phonies.  That's his biggest complaint of all.  

"One of the biggest reasons I left Elkton Hills was because I was surrounded by phonies. That's all. They were coming in the goddamn window. "

Later, Holden tries to explain to Phoebe why he got kicked out of Pency. 

"A million reasons why. It was one of the worst schools I ever went to. It was full of phonies."

To Holden, a phony is anybody that puts on an act and has become so good at it that they don't even realize they are doing it.  And I see Holden as a huge phony.  

Throughout the novel he is all bluff and bluster.  He talks a really big game about leaving and going somewhere else.  He talks big about how everybody around him are all such phonies and about how he can't stand it anymore and is going to leave.  

He even manages to convince his sister to run away with him.  But at the end of the story, when his sister is ready to go, Holden changes his mind and decides to stay.  At that point, I consider him a phony along with all of the others.  

He decides to finally make a change in his life and do something proactive, and then he chickens out (that's how I read it anyway). I would have been much more satisfied if Holden had actually followed through on his plan.  As it is, the ending fits Holden's personality.  So, on one hand, the ending makes perfect sense. On the other hand, it didn't satisfy anything for me.