If a card or ability instructs your opponent to put cards into their hand without specifically using the word "draw," it's not a card drawn.

The last ability can trigger only once each turn for each opponent. It doesn't matter whether The Unagi of Kyoshi Island 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.

If an effect instructs your opponent to draw multiple cards, the ability triggers after they draw whichever is the second one for the turn. It goes on the stack after they've drawn and looked at all the cards and finished resolving the spell or ability that caused them to draw cards.