If you cast a spell “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, such as that of Swallow Whole, those must be paid to cast the card.

If you wish to cast the target card, you do so as part of the resolution of Vadrok’s ability. You can’t wait to cast it later. Timing permissions based on a card’s type are ignored.

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