Upon making a legal serve over the net in table tennis, can the ball hit the receiver's side more than once? And, if so can it spin back and hit the net on the receiver's side?
The answer is YES to both of your questions.
In fact skilled players will mostly want their service to drop shortly after the net, so that the opponent has a hard time to attack it. Serving with such backspin that the ball returns to your side after one touch on the other side is really really hard. Even after a lot of practice you will only achieve this for very high services which the opponent can smash.
If you want to practice short backspin serves, you can assess your progress by counting the balls that return towards the net vs. those that drop off the table behind the opponent's base line.
For the first question: Upon making a legal serve over the net in table tennis, can the ball hit the receiver's side more than once?
The answer is yes! It's no matter the long, and short serve, the serve is legal. But if the ball hit the receiver more than once, then the receiver lose the point.
For the second question: And, if so can it spin back and hit the net on the receiver's side? Based on the basic serving rules, if the ball spin back to the server's side, then the server get the points.