Does the reciever need to wait for the ball to bounce on the table before hitting it back to the opponent?
My question refers to both singles and doubles.
Yes, the receiver needs to wait until the ball touches his playing area, otherwise it counts as being obstructed.
You can read about it in table tennis rules
2.05.08 A player obstructs the ball if he or she, or anything he or she wears or carries, touches it in play when it is above or travelling towards the playing surface, not having touched his or her court since last being struck by his or her opponent.
And a player win a point in case his opponent obstructs the ball:
2.10 A Point
2.10.1 Unless the rally is a let, a player shall score a point
2.10.01.05 if an opponent obstructs the ball;
Yes, absolutely - the receiving player must allow the ball to bounce before returning the serve - or any shot, actually. If that was not required by the rules, it would be very easy for the returning player to hit (volley) the ball right as it crossed over the net with a shot that would be nonreturnable by the server, and points would be very short. Basically, it would provide an unfair advantage.