No. Outside of the battlefield the back face of a DFC doesn't exist for anything that cares about it. The reason Commander can "see" them is because there is a specific rule allowing it to break convention. Anything else will only see the front face if they are not on the battlefield.
903.4c. The back face of a double-faced card (see rule 711) is included when determining a card's color identity. This is an exception to rule 711.4a.
No, only the flying and vigilance abilities on Bruna are relevant.
712.3a While a meld card is outside the game, in a zone other than the battlefield, or on the battlefield with its front face up, it has only the characteristics of its front face.
"Color identity" matters for EDH deck construction (and certaincards) because it is explicitly referenced there. "Ability identity" neither exists nor is referenced on any card.
Private Mod Note
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Rollback Post to RevisionRollBack
Why bother with mere rulings when so many answers can be found in the Rules?
No, only the flying and vigilance abilities on Bruna are relevant.
"Color identity" matters for EDH deck construction (and certain cards) because it is explicitly referenced there. "Ability identity" neither exists nor is referenced on any card.