Why? Tywin outlived Joffrey, after all.
Of course they are. Even if Robb isn't explicitly aware of Stannis declaring himself king, he's aware that Renly has and knows Stannis has the stronger legal claim (And later he learns of Stannis' claim anyway); the decision to keep fighting for independence is made anyway.
The Lannisters were willing to make peace; they (Tyrion, specifically, if I remember right) assumed that the execution of Ned would make it impossible. Robb's unacceptable peace terms probably only confirmed it, but if he'd offered reasonable ones, you don't think Tywin would have accepted to focus on Renly (and then Stannis)?
I don't think that's really accurate, since Catelyn advocates for peace even after Ned's execution; if she was willing to put her feelings aside, clearly a peace was possible. The chances for a peace were more undermined by
(1) Failure to form a Stark/Baratheon (either Renly or Stannis) alliance
(2) Renly's death leading to the Tyrell/Lannister alliance
(3) Robb's intentionally unacceptable peace terms
As time goes on, the southern picture clarifies itself outside of Robb's control and, unsurprisingly, he pulls the short straw.
The desire/need for vengeance is completely understandable, but it was a self-destructive motivation.
Edited by Ser Hippie, 12 April 2012 - 11:21 AM.