If an effect instructs you to draw multiple cards, Niv-Mizzet's first triggered ability triggers that many times. You choose targets for those abilities after you've drawn all of the cards.

If a spell or ability causes you to put cards into your hand without specifically using the word "draw," Niv-Mizzet's first triggered ability won't trigger.

Niv-Mizzet's second triggered ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered. This causes its first triggered ability to trigger, and that also resolves before the spell.

Players can cast spells and activate abilities after Niv-Mizzet's second triggered ability resolves but before the spell that caused it to trigger does. Notably, the card you draw may be able to counter that spell.