If a spell or ability causes you to put cards into your hand without specifically using the word “draw,” it’s not a card drawn.

If an effect instructs you to draw multiple cards, Jace’s first ability triggers after you draw whichever is the second one for the turn (if any). You choose a target for the ability after you’ve drawn all of the cards.

Once Jace’s last ability has resolved, its effect applies even if Jace has left the battlefield.

Jace’s first ability can trigger only once each turn. It doesn’t matter whether Jace was on the battlefield when the first card was drawn.