What has Gene learned about hatred, jealousy, friendship and war?
Gene experienced each of these four things; what or how much he learned will be up to you to decide.
Hatred - Hatred is a form of jealousy, which is dealt with below. The only hatred Gene had was short-lived, but it had lasting ramifications each time it appeared. In the moment when he felt hatred for Finny's trying to subvert his academics, he jounced the limb. When he felt hatred for Leper's accusations that Gene hurt Finny intentionally (which of course he did), Gene got physical with him. Finally, when Gene is faced with the truth of his actions in front of Finny in Brinker's mock trial, more hatred was shown, but here the damage happens to Finny. Gene learned hatred is a costly emotion.
Jealousy - Gene was always a little jealous of Finny's natural athletic and social abilities. At one point, when that jealousy flared, Gene committed an impulsive, cruel action which did irreparable harm to Finny. He regretted it soon after, and he learnedthat Finny was a genuine friend. Gene learned that jealousy, when acted upon, can be deadly--literally.
Friendship - This is the major conflict in this novel. Gene at some point believedthe rather happy-go-lucky Finny was trying to sabotage Gene's educational pursuits in order to make himself look or feel better. He was so sure, for that short time, that this was Finny's plan that he jounced the limb and effectively ended Finny's life. When Gene realized Finny was, indeed, a true and faithful friend, it was too late. Gene learned the value of a true friend and the dangers of jumping to conclusions and placing his own motives onto others.
War - The war was a steady drumbeat in the background of this entire story. It is real and Gene and his classmates all know it--even Finny, who appears to be in denial--and spend their time preparing for it. When Leper enlists, The boys all get a jolt of reality; but when Gene visits the AWOL Leper at his home, Gene sees first-hand the potential damaging effects of war. Toward the end of the story, he and Brinker plan to just enlist and get it all over with. Instead, Finny returns and those plans are dropped. At the end of the year, when Brinker's dad tried to convince the boys that "doing their duty" was a great and noble thing, both boys believed it was neither great nor noble to serve. Gene spent a year in the service, but he never saw battle. War was one of the growing up events of this novel, and it's part of Gene's disillusionment.
You are right to identify that this is a novel where the main protagonist learns something, for in many ways, this novel can be considered to be a bildungsroman, or a novel of education, where the central character embarks on a learning journey that makes him a wiser individual by the end of the story. Remember, you are only allowed to ask one question, so instead of responding to each of the very valid and important themes that you have highlighted, I will talk about the theme of war in the novel and what Gene has learned about it.
I think the crucial part of the text to examine with regard to the theme of war is actually the end of the last chapter. Consider what Gene confesses about his involvement in the war:
I never killed anybody and I never developed an intense level of hatred for the enemy. Because my war ended before I ever put on a uniform; I was on active duty all my time at school; I killed my enemy there.
This quote clearly points towards an expanded definition of what we consider to be "war" - Gene is pointing towards the fact that in some senses our lives can be considered to be lived in wartime, whether or not there is an actual war going on. In many senses, the war in A Separate Peace provides a visual backdrop to what is going on in the hearts and minds of the characters, as Gene goes on to state:
Other people experienced this fearful shock somewhere, this sighting of the enemy, and so began an obsessive labor of defense, began to parry the menace they saw facing them by developing a particular frame of mind...
However, as the novel concludes, this state of mind is ultimately futile and perhaps even damaging to the person themselves as shown by the other characters and their response to war:
All of them, all except Phineas, constructed at infinite cost to themselves these Maginot Lines against this enemy they thought they saw across the frontier, this enemy who never attacked that way - if he ever attacked at all; if he was indeed the enemy.
Perhaps, Gene suggests, as he has learned, that the real enemy isn't who we suppose the real enemy is, but is in fact within you - that is the real "war" that is carried on and that we fight.
Gene certainly learns about friendship and jealousy through his own experiences.
First of all, from the get-go, he is jealous of but delighted by Finny. He wants to do anything he can to impress Finny at first because he is just completely attracted to his character. Often in new relationships we all feel that way - there are just certain people who intrinsically attract our attention and devotion. Gene wants to perform well at Blitzball and not give up to impress Finny. He compares every aspect of his size and appearance to Finny. He notes their similarities with great fondness.
However, by chapter 2, Gene is already beginning to experience jealousy as he watches Finny thrive. The Headmaster's Tea demonstrates Finny's verbal-linguistic ability to convince even teachers that what he does wrong is actually right. Gene at that point vows to increase his efforts in his studies because that is an area of competition that he can certainly exceed far beyond Finny. The two do eventually build dependence on each other as they value each others strengths, but as with most things for guys, they rarely admit this.
Gene learns that friendship requires trust. He doesn't ever build the relationship he wants with Finny because of the guilt he feels for never totally ensuring that Finny understands the truth of the incident at the tree. Because Finny finds out about this late, he loses faith in Gene and Gene ends up feeling even more horrible as a second break of Finny's leg feels like his fault again. Whether the lesson really hit home as a teen we don't really know. But because he is telling this entire story as a flashback, the adult Gene seems to have learned from the experience.
In the end, Gene looks back on his years and experiences at Devon and realizes he has learned that time resolves, or at least diminishes, all the emotions and situations that seem so momentous when one is immediately involved in them. All the conflict and rivalry and envy and affection and admiration he felt for Finny; all the enthusiasm and boredom and doubt and apprehension and apathy he felt toward the war; none of it lasted.
When Gene visited Devon fifteen years after his graduation, he was able to locate important buildings and landmarks that had been critically important to his student activities. However, he found they
seemed...to resemble those men, the giants of your childhood, whom you encounter years later and find that they are not merely smaller in relation to your growth, but that they are absolutely smaller...the old giants have become pigmies while you were looking the other way...Nothing endures, not a tree, not love, not even a death by violence.
At last, Gene realized he had made his own peace with the events of that period in his life.