If you cast a card “without paying its mana cost,” you can’t pay any alternative costs. You can, however, pay additional costs. If the card has any mandatory additional costs, those must be paid to cast the card.

The last triggered ability targets the opponent. If that player is an illegal target as the ability tries to resolve, the ability won’t resolve and none of its effects will happen. You won’t exile a card.

If the card has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

The two cards you put on top of your library can be from the three you just drew or ones that were already in your hand.

You choose whether or not to cast the instant or sorcery card as the last triggered ability resolves. If you do, you do so as part of the resolution of that ability. You can’t wait to cast it later in the turn. Timing restrictions based on the card’s type are ignored.