QIf I discard 2 cards at the same time, how many times does this follower's automatic ability trigger?

AIt would trigger twice, with both instances becoming pending and resolving in the order of your choice. However, once you pay the cost of the ability by engaging this follower, you would become unable to pay the cost for subsequent instances, meaning you would only draw 1 card total.

QIf I discard a card during an opponent's turn, does this follower's automatic ability trigger?

AYes, it does.