Guys guys guys.
We are NEGOTIATING.
We have the advantage here, but it would be out of character for Ned to abuse it and pushing to hard will result in Casterly Rock going to war.
We have to be able to justify our decisions legally at least.
Tywin, as far as I can remember, is guilty of nothing more than believing his grandson was the legitimate king. He's super dangerous, but we can't really justify more than ransom.
Jaime IS guilty of incest, cuckolding the king, and putting a bastard on the throne. Execution or taking the Black.
Cersei is the same as Jaime plus possible regicide. Unfortunately, she's the one we are negotiating with. Best we can hope is she joins a convent.
Tyrion is similar to his dad as far as we are concerned. Ransom.
Cersei's children are guilty of nothing except disgusting parentage. Joffrey is a psychopathic monster, but has done nothing but try and take what he believed to be his rightful thrown. And they are the most important bargaining chips. I think our best bet is to make them wards of various lords.
Gregor Clegane. A monster and criminal. Technically we could offer him the Black but it would piss off the Riverlands. Execution. Possibly in Dorne.