The Company of Myself/WMG

Jack killed Kathryn over something petty.
Like completing a level in a computer game (After all, everything that happens in the game is a representation of what went through(and is still going) Jack's mind.

Alternatively, Jack had a... well, not a good reason, but a better reason for killing Kathryn...
She didn't really love him as much as he loved her. Notice the levels in which they cooperate. Kathryn can stand on his head as leverage, but the opposite is not possible. You can argue that this is simply because Jack is physically stronger and is just able to support her weight, but since everything else in the game is actually a metaphor for something, maybe that means that their relationship was very one sided. Maybe she even cheated on him. So he killed her.

Jack didn't kill her.
Well, not on purpose. Whatever happened, it was a genuine accident. There was no malicious intent. He didn't mean to hurt her in any way, but he blames himself for her death regardless. Maybe he confessed to her "murder" due to a sense of guilt, believing that he should be punished for her death no matter why it happened.
 * Unlikely. In Fixation, when they reach that area, he claims "the path to the right leads back home" and then promptly takes the left, actually homeward path as soon as it opens up. He seems to know all too well what he's doing.
 * Fixation also implies that he wasn't acting normally, however -- he says early on that his meds make him do things he normally wouldn't.