I see Theon dying to help his true family, via Rickon or Sansa, maybe Bran.
Jaime is definitely surviving his run-in with LSH. I'm worried about Brienne, too, but I think the BwB will force her into some kind of dilemma/morally difficult quest. Her character and AFFC/ADWD arc is too close to the core idea of the story (keeping to a moral in the face of hardship) right as that idea is being put to its final test.
I think her worst/most violent actions will take place in Yunkai and Pentos, what with Yunkai being the chief traitors/opponents and Illyrio manipulating her from the beginning. And both cities best represent the corruption and ill-gotten wealth of Essos, what with Astapor already fallen and Volantis having a majority slave population (and the city has been from the populist/servant side more than the elitist/master - our POVs walk the streets and see the strife, and only hear about the Old Blood secluded behind the Black Walls). Volantis will fall to a slave uprising. It'll be as bloody as Astapor, but maybe not as much of a mark on Dany's character.
Not counting them because of mental illness is a copout. As far as we can tell, Gregor suffers from gigantism and migraines, which causes him a great deal of pain that he needs to medicate. This isn't a reason to murder and rape. And with all of these characters, GRRM offers histories that make it clear the influence of whatever conditions they have was channelled through circumstance: however much Gregor's headaches made him what he is, he was equally shaped by his upbringing, which in turn was affected by his physical size, which changed his position in society. Highly valued as a soldier, relied on by his family to improve their station, rewarded for killing people, excused for doing so brutally on account of his worth. They become who they are as a result of many factors, most out of their control, but this is the case for all characters.
Yes, they were raised with worse teachings than, say, Robb and Jon. But patterns can be broken and upbringing can be overcome. It's difficult, and that difficulty is what makes some dubious characters capable of heroism when they succeed: Tyrion, Theon, Jaime, and so on. Davos was born to a difficult position that lead him to a life of crime. And then he risked his life to save Stannis and his people, and went on to be good by doing good, breaking rules and risking his life to save others and stop evildoers, or even just to speak out against them, all while being discouraged by his surroundings and told to accept, condone, imitate.
Ramsay, meanwhile, continues to prove himself monstrously evil. We can weep for what the child Ramsay could have been, but Ramsay the man has bought into everything he was taught. Yeah, he's unusually cruel, but so is Roose. I don't think Ramsay is meant to be viewed as a psychotic anomaly, but the worst example of something common, otherwise exhibited by Roose, Tywin, Littlefinger, etc.