Mind's Dilation
Oracle Text
Whenever an opponent casts their first spell each turn, that player exiles the top card of their library. If it's a nonland card, you may cast it without paying its mana cost.
Card Rulings
7/13/2016 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.
7/13/2016 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.
7/13/2016 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.
7/13/2016 If you cast the card this way, it will resolve before the spell that caused Mind’s Dilation’s ability to trigger.
7/13/2016 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.
7/13/2016 If you don’t cast the card, perhaps because you choose not to or because it’s a land card, it remains exiled.
Or is it the type thing where the revealed card sits in exile until whenever the mind's dilation player can/wants to cast it
While the above three examples allow a wider window in which to cast the exiled cards, the casting of the exiled card(s) must follow normal timing restrictions. In the case of Mind's Dilation however, timing restrictions are ignored. MtG has a storied history of ignoring timing restrictions when casting cards from exile upon singular triggered abilities such as Cascade, Madness, and Rebound.
In the case of your Rout example, if you reveal a creature you want, you may cast it, putting it on the stack, then cast Rout for its instant speed cost, putting it on the stack above the exiled card you have cast. Finally, Rout resolves, wiping the board, the exiled creature spell resolves, entering the battlefield under your control, then lastly, your opponents spell which triggered Mind's Dilation will resolve.
--> You would have to cast Rout before the trigger resolves, otherwise your new creature is doomed like everything else.
Edit: forgot you can cast Rout while this is on the stack...