If the bat slips from the batsman's hands, and the ball touches the bat, and is then caught by a fielder, should the batsman be given out?
No - the batsman can only be out if he hits the ball with the bat, or a hand holding the bat, whilst in the commission of a stroke. If he is not holding the bat, and the ball strikes the bat, it's not out. See Law 6.8 "The bat"1
Depending on the circumstances, it could be obstructing the field, for instance if he throws the bat at the ball to prevent it striking the stumps for a runout. This would need to be a deliberate act.
1. After the Laws were renumbered in 2017, this is now Law 5.6.