What happens if a pitched ball bounces and the batter hits it?

The answer is listed under 2.00 - "Definitions of Terms" official Baseball rules. It states that: If a pitch touches the ground and bounces through the strike zone it is a "ball". If such a pitch touches the batter, he shall be awarded first base. If the batter hits such a pitch, the ensuing action shall be the same as if he hit the ball in flight.

Yes. MLB Rule 6.08(b) makes no distinction as to whether the ball does or does not hit the ground prior to hitting the batter. MLB Rule 6.08(b) is below:

" He is touched by a pitched ball which he is not attempting to hit unless (1) The ball is in the strike zone when it touches the batter, or (2) The batter makes no attempt to avoid being touched by the ball; If the ball is in the strike zone when it touches the batter, it shall be called a strike, whether or not the batter tries to avoid the ball. If the ball is outside the strike zone when it touches the batter, it shall be called a ball if he makes no attempt to avoid being touched.

APPROVED RULING: When the batter is touched by a pitched ball which does not entitle him to first base, the ball is dead and no runner may advance. "