This question is specifically in relation to Mastermind's Acquisition's wish option to "Choose a card you own from outside the game and put it into your hand."
If I were to own a card that has been exiled face-down to some effect like Necropotence, Thief of Sanity, Praetor's grasp etc. would I be able to wish a card exiled that way into my hand?
Additionally if I had multiple cards exiled facedown, would I be able to select a card from among them?
In a sanctioned format you must choose a card from your sideboard. In casual play you can choose any card, but it must be from outside the game which means it cannot be in exile.
This question is specifically in relation to Mastermind's Acquisition's wish option to "Choose a card you own from outside the game and put it into your hand."
If I were to own a card that has been exiled face-down to some effect like Necropotence, Thief of Sanity, Praetor's grasp etc. would I be able to wish a card exiled that way into my hand?
No. A card in exile is not "outside the game".
400.1. A zone is a place where objects can be during a game. There are normally seven zones: library, hand, battlefield, graveyard, stack, exile, and command. Some older cards also use the ante zone. Each player has their own library, hand, and graveyard. The other zones are shared by all players.
400.10. An object is outside the game if it isn’t in any of the game’s zones. Outside the game is not a zone.
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Why bother with mere rulings when so many answers can be found in the Rules?
Is a card that is exiled not "removed from the game"
No, when magic was first created they had terminology 'remove from the game' which eventually turned into the exile zone. You used to be able to use wishes for cards when this terminology was used but it was changed when people used it for infinite combos with Burning Wish.
Even if the spell or ability allowed you to choose a card "you own ... in exile" (e.g., Coax from the Blind Eternities speaks of an "Eldrazi card you own from outside the game or in exile"), the card would still have to meet the criteria of that spell or ability. Note, however, that in general, a face-down card in exile has no characteristics, even if—
the card would meet the spell's or ability's criteria if it were face up, or
something allows one or more players to look at that card in exile
(C.R. 406.3a).
Thus, for example, in general, a face-down card in exile isn't an "Eldrazi card" for the purposes of Coax from the Blind Eternities, since the face-down card has no subtypes, including Eldrazi (C.R. 203.5m, 109.3).
EDIT (Feb. 4, 2021): Edited to conform to rule update for Kaldheim.
If I were to own a card that has been exiled face-down to some effect like Necropotence, Thief of Sanity, Praetor's grasp etc. would I be able to wish a card exiled that way into my hand?
Additionally if I had multiple cards exiled facedown, would I be able to select a card from among them?
No. A card in exile is not "outside the game".
No, when magic was first created they had terminology 'remove from the game' which eventually turned into the exile zone. You used to be able to use wishes for cards when this terminology was used but it was changed when people used it for infinite combos with Burning Wish.
Thus, for example, in general, a face-down card in exile isn't an "Eldrazi card" for the purposes of Coax from the Blind Eternities, since the face-down card has no subtypes, including Eldrazi (C.R. 203.5m, 109.3).
EDIT (Feb. 4, 2021): Edited to conform to rule update for Kaldheim.