TL;DR!!!
Why James finds Azula so fascinating
Well, it was simple curiosity at first, but he sees her as a challenge and the whole Peter thing makes him want to enjoy life in case he can't change the future and he wants to help others do the same.
It's also stubbornness. She's asked him to leave her alone and he did, and then her birthday turned up and she still hated him. So now he's trying to get her to open up to an Anon so he help without her knowing it's him and hopefully win her friendship. Because he only wants friendship, nothing more.
Well, it was simple curiosity at first, but he sees her as a challenge and the whole Peter thing makes him want to enjoy life in case he can't change the future and he wants to help others do the same.
It's also stubbornness. She's asked him to leave her alone and he did, and then her birthday turned up and she still hated him. So now he's trying to get her to open up to an Anon so he help without her knowing it's him and hopefully win her friendship. Because he only wants friendship, nothing more.