The triggered ability can trigger only once each turn. It doesn’t matter Gavi was on the battlefield when the first card was drawn. If it’s not on the battlefield when the second card is drawn, the ability can’t trigger at all that turn. It won’t trigger when the third or fourth card is drawn.

Gavi’s first ability considers the entire turn. If you cycle a card then have Gavi come under your control, you can’t pay {0} to cycle your second card.

If a cycling cost includes {X} and you pay an alternative activation cost of {0} instead, X is 0.

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.