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 section 8 of the relevant law - http://www.lords.org/mcc/laws-of-cricket/laws/law-6-the-bat/
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.