No, you cannot play a face-down Misthollow Griffin from exile. The reason is that "face down" is a status of an object in the game. Even if, in practice, you happen to turn the physical card face-up, the game still considers it a face-down object.
This is important because a face-down object has no characteristics, and in particular, no abilities - it's not really a "Misthollow Griffin" object, it's just a plain old "face-down featureless card" object. (CR 406.3)
406.3. ...A card exiled face down has no characteristics, but the spell or ability that exiled it may allow it to be played from exile....
The fact that one or more players are allowed to look at the card doesn't change the fact that it's considered a face-down object.
That also addresses your second question: no matter who can look at the card, it's still considered a face-down card object with no characteristics, so it doesn't have the ability that allows it to be played from exile.