If you cast the exiled card, you do so as part of the resolution of Mind's Dilation's ability. You can't wait to cast it later in the turn. Timing permissions based on the card's type are ignored, but other restrictions (such as "Cast [this card] only during combat") are not.

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

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

If you cast the card this way, it will resolve before the spell that caused Mind's Dilation's ability to trigger.

If you cast an instant or sorcery card this way, it goes to its owner's graveyard as normal. It doesn't return to exile.

If you don't cast the card, perhaps because you choose not to or because it's a land card, it remains exiled.