QIf my opponent's deck is empty, and I make them put the top card of their deck into their cemetery, do I win the game?

ANo. If you draw a card when your deck is empty, you lose the game. If you put the top card of your deck into your cemetery when your deck is empty, nothing happens.

QIf my opponent puts 2 cards from their deck into their cemetery during my turn, does this follower's ability trigger 2 times?

AYes. It triggers 2 times, so you give this follower +2 attack.