If you cast the card, you do so as part of the resolution of Torrential Gearhulk's triggered ability. You can't wait to cast it later in the turn. Timing restrictions (such as "Cast [this card] only during combat") still apply.

If you cast a card "without paying its mana cost," you can't pay any alternative costs, such as emerge costs. You can, however, pay additional costs. If the card has any mandatory additional costs, such as that of Incendiary Sabotage, you must pay those to cast the 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.