You don’t choose the exiled cards until Mirror of Fate’s ability resolves. It doesn’t matter how they wound up in the exile zone.

Exiled cards are face up by default; they’re face down only if the effect that exiled them said so. For example, the cards from your library that Mirror of Fate exiles are exiled face up.

You choose the order that the chosen cards are put on top of your library. You don’t have to show anyone else that order.