Bumping. Setting. And spiking.
The only exception I believe is when a blocker has blocked the ball and they can also pass it if they get the ball in time after the block. But that is the only time someone can hit the ball twice in a row.
The ball is still technically in. According to the rules of volleyball, if it hits the net and goes over, it is still in.
The team that hit the ball over will get a point.
Even of the ball hits the top of the net, if it goes over, it is still in play.
Setting in volleyball is when a player hits the ball over the net in a forward pass motion. The ball is hit off the finger tips and not spiked with the fist.
Nothing, this is completely legal.