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 Thrill of Possibility, those must be paid to cast the card.

If your opponent wishes to cast the card, they do so as part of the resolution of Allure of the Unknown. They can’t wait to cast it later. Timing permissions based on a card’s type are ignored.

If all six of the revealed cards are land cards, your opponent doesn’t exile anything. You put all six cards into your hand.

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