We see that question here quite often. It usually takes quite a bit of convincing before people understand that it's legal.
>>> If so, any time a dribble is stopped, one can pass the ball off the board and legally move?
Yes, he can, as long as the backboard is his own backboard. If you throw it off the opponent's backboard, it's the same as a dribble. Such is not the case when you throw it off your own backboard.
>>>Can they start another dribble?
Yes, of course. He can also shoot, pass, ask for a timeout, or throw the ball off the backboard again.
Instead of being curious why it's legal, explain why it should be illegal, with rule references please.