Can the batsman hit the ball after the ball crossed the stumps?
It is allowed. There is no rule for this.
Brad Haddin once(two times same match) did this on a free hit. Just because you can't get bowled on a free hit and you get more time when you play after the ball passes the stumps.- Video Link.
This has hardly happened in International cricket.
Sometimes Upper cuts, unintentionally are hit after the ball crosses the stumps.