QMy opponent has Ancient Protector on their field, and I have Spartacus. While resolving this follower's "at the start of your end phase" ability, I draw enough cards that my deck is down to a single card. Do I then win the game?

ANo, the game goes on. When one effect instructs a player to perform an action ("win the game") that is prohibited by another active effect ("you can't lose the game, and opponents can't win"), the prohibiting effect takes precedence.

QIf my deck is empty when playing this follower's "at the start of your end phase" ability, do I lose the game?

ANo, you win. After discarding any number of cards, you draw that many plus 1. Drawing cards from an empty deck means you will lose the game at the next confirmation timing. However, you finish resolving this ability first, and since there's 1 card or less in your deck, you win the game immediately.

QCan I choose not to discard any cards when playing this follower's "at the start of your end phase" ability?

AYes. "Any number" includes 0. If you choose not to discard any cards, you draw one card.