QIf I begin my turn with 0 cards in hand, then draw 1 during my start phase, do I then have to discard it with this follower's ability?

AYes. Each turn consists of a start phase, main phase, and end phase. After drawing a card in your start phase, your main phase begins, triggering this follower's ability.

QCan I choose not to discard a card at the start of my main phase?

ANo. You must discard a card if you can.