Why by the old gods is Robb going to war to begin with?

[James Arryn]

First, the progression as seen from the Stark POV: Bran pushed from window. Another attempt on Bran's life along with Cat being attacked. Robert dies, Ned declared traitor, his men killed, he imprisoned. That's the starting position. So, any leverage they had to get Ned back based on the status quo had already proven insufficient. The only way to change that was to gain leverage somehow. And in order for Robb to be taken seriously as a leader...ie, be in a position to gain any leverage...he had to act. Inaction in the face of attack is like folding all night long; you're assured of a slow loss. So he had to try and force the crown's hand, and rally his men. Interests had a mutual solution; do what your father did in a not completely different situation. Call the banners, march south, make the juice not worth the squeeze for the Lannisters. Then they kill Ned and the RL align themselves with you. It might be tiger by the tail time, but I maintain that he was correct in continuing at this point. The game changer for him was Balon being insane...but that's a chance you have to take. If you wait it out and the Lannisters prove able to deal with the Baratheons in turn...assuming your men haven't abandoned you by this point for showing serious weakness....then you are vulnerable to a concentrated multi-front attack qith questionable support and the likelihood that one of your lord decides that you aren't worth the cost when you didn't even back your own play.

[The Frosted King]

I swear it's the same suspects dropping poor and altogether unsatisfactory alternatives.

You know who you are.