Do you get points when your opponent serves in ping-pong?

Answer 1

Yes.

You win points if you keep the rally going longer than your opponent.

It doesn't matter who is serving. It's not like Badminton or squash where you only score points on your own serve.

Answer 2

The order of service is maintained at game point but if the game is tied at 10 each, players only serve once rather than serving twice.