Given the below trope, it's possible that Judy has gone into shock after Buzz's sudden death and her Anguished Declaration of Love (as well as a couple of other bits of odd behaviour) are all side effects of that.
Plato's fascination with Jim. There's a lot of Ho Yay involved, so many fans take him to be Ambiguously Gay. The way he talks to Judy about him sounds like one would talk about someone they were in love with. But then again, a major part of his angst comes from not having a father - and in their interactions in the 'castle', Judy and Jim act as surrogate parents to him. With the noticeable age difference between them, his interest could entirely be platonic (and that's in his chosen nickname) and he just wants a father figure. Of course there's no reason it couldn't be both too.
Angst? What Angst?: Judy takes her boyfriend's death pretty easily. She's shown trying to hold it together when she comes home late, but it's practically forgotten the next time we see her.
Ensemble Darkhorse: Jim and Judy are the first ranking protagonists, but a lot of viewers feel that Plato is the true protagonist of the story. Even the director agreed.
Foe Yay: The legendary exchange between Buzz and Jim:
Ho Yay Shipping: Jim and Plato. Of course, considering that James Dean was rumoured to be bisexual, director Nicholas Ray was definitely a bisexual and that Sal Mineo was also bisexual, it's more than just subtext.
This film made or codified several tropes that went on to be mainstays of the way teen angst would be depicted for the next 50 years. For example, elements of several early episodes of The O.C. are explicitly drawn from parts of this movie. As a result, much of it seems cliché today; when it was released, it shocked audiences.
It should also be noted that the film is not about "rebelling" as such but having "nothing" to rebel against, in that the cast are spoiled rich kids with everything provided for them but not really content and seeking a deeper purpose and a community and culture of their own. This was a huge subversion of the juvenile delinquent theme which believed Society Is to Blame. The point is that post-war prosperity will not alone guarantee safety and comfort.
Also the first film which showed teenagers living and forming a counter-culture which heralded the 60s in a big way.
At one point Jim realizes he's so screwed up because his dad never beat his mom. Jim thinks his father is weak because he seems to do a lot of the cooking and cleaning in the house, and Jim believes that cleaning up is women's work. Today, any man that doesn't at least clean up after himself is likely to find himself divorced and out on his ass.
Plato's Mammy doesn't bat an eyelid at the boy having a gun in the house, or try to confiscate it after he's taken to the police station for shooting puppies.
Writer-Induced Fanon: Since this is The '50s, Plato is not specifically identified as gay in the film, but he was intended as such by the director and actors, who confirmed the hints that critics and audiences had picked up since the film's release.
The Woobie: The three protagonists to some extent. Jim has a difficult relationship with his parents, especially his father, and just seems so lost. Judy's relationship with her father is strained too, and it's implied that he ignores her a lot. Plato of course is introduced with no friends at all on his birthday, and is so lonely.