Well the books make it clear that Jaime truly loves his sister. I think there's hints of that in the show, but maybe not enough for someone who hasn't read the books to pick up on them. (I have no idea if you've read them or not, it's merely a general statement.)
That may be true from the books; there are always changes in adaptations and it seems like this is an area where they may have subtly shifted things.
I agree that Jaime is pretty far gone as a human being. But this isn't to say he's beyond redemption. It just means he has a long way to go. A hell of a long way. Brienne might get him there.
I did get the sense that Brienne was supposed to be sort of his way out. But he didn't want it. He stuck with his sister, who besides being his sister is a horrible person. He still wants to be accepted by his father too. Those inextricable relationships seem like they'll be the end of him, as both of those people are using him to serve their own ends.
I admit I hadn't thought about the connections between PTSD and rape.
To me, the scene where he first gets back and sees Cersei and she rejects him brought up analogous real-world scenes that I've seen. He's a man who's waited a long time for that moment. He probably was imagining it the entire time he was chained up; it probably it what got him through his ordeal. He was banking on their reunion being the thing that healed all the pain. To me, there's been a simmering rage beneath his cool exterior ever since then.
To be honest I don't think of any of the Lannisters as psychopaths. Bad people. Very, very bad people. But in all their cases there's some redeeming features and believable reasons for how they came to be who they are.
As is the case with most psychopaths. Psychopathy is a personality trait; there are degrees. It also runs in families, and it seems to me that the message the show is sending is that Joffrey is what happens when you combine two people with antisocial tendencies: you get someone twice as bad.
Many of the worst people we think of being associated with that word went through childhood trauma. There's also increasing evidence that a degree of psychopathy is very useful in getting to the highest levels of the business world; we're not just talking about violent criminals. To me, the whole Lannister family exhibits various degrees of it (and, indeed, so do all the power players in King's Landing); and there are things that explain why but they don't negate that truth.
In Jaime's case it feels to me like his most moral act was killing the old Targaryen king. It's clear that he feels he did it not for selfish reasons to advance his own family, but because the king was a horrible person. To him, it was worth murdering someone and breaking his knight's oath for the greater good. But no one else saw it that way; even his own family doesn't seem to think much of him once he's done such a dishonorable thing. To me, the lesson he learned from that is that no good deed goes unpunished, and he feels very jaded and spiteful throughout the series.
I also suspect that he sees the parallels between the old king and the new one lying in front of him; and despite that king being his son I don't think he is as blindly in love with him as Cersei is. He's probably okay with Joffrey being dead and realizes that the king was a monstrous one, and the fact that Cersei seems to love her tyrant son more than either of her less evil brothers (as evidenced by asking one to kill the other over this) is probably eating him up inside. Jaime may have been the best of all of his family at one point, but I think the show has amply established what's rotting him inside.