Can a batter, after a foul tip on strike two, run for first base since it is a live ball? What happens if he stays at homeplate?.
No, the batter cannot run. The ball is "live" for the purposes of the defense and any runners already on base, but it doesn't allow the batter to leave, and the batter is not in jeopardy of being put out.
This situation (foul tip on strike two) does not fit any of the criteria of 5.05, and therefore does not entitle the batter to become a runner.
The batter may only advance to first base on a dropped third strike. A foul tip may never become a dropped third strike, as if it is dropped it is treated as a regular foul ball (and thus is not in play), and if it is caught it is not dropped (and thus the batter is simply out).
The runner already on base may steal on a foul tip since he is already on base, and thus may treat a foul tip like any other pitch (the ball is in play). But the batter would be out as soon as the foul tip was caught and therefore could not advance.