And the war started the second Robert dies and Catelyn had nothing to do with that either.
As soon as it was known that Catelyn had kidnapped Tyrion, Tywin sends Gregor Clegane to burn and destroy the Riverlands, while he gathers his army. Ned, while Robert is still alive and hunting, then sends Beric Dondarrion to kill Gregor, and demands Tywin answer for Gregor's crimes or be branded a traitor. Tywin has assembled his army and gives Jaime half the army to attack Riverrun, Catelyn's girlhood home. I'd say war has already begun at this point. Then after all that King Robert is gored by a boar.
She kidnapped Tyrion because she was led to believe that he tried to kill her son.
Before she even left, Maester Luwin cautioned everyone about acting hastily. Her husband told her not to do anything rash. Everyone told her there wasn't enough evidence, and making accusations against the Queen's brother could be considered treason. But she went ahead and ignored all the sensible advice, and disobeyed her husband. Then to top off her stupidity she takes him to her sister; the only thing that might have been legal would be to take him before the king.
I think you are wrong about Catelyn.
She doesn't hate John, she resents him because he is a reminder of Ned's cheating.
The show has to skip lots of small scenes that might give you more evidence of how much she despises Jon, but the book gives a fuller picture. Also I think the shows wants her to be a little more sympathetic than she is in the books.
If the show gets to season 3, you'll see how strongly she hates Jon.