原文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.
原文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.