The problem I see is that the "while maintaining possession of the ball" [in my mind] is meant to mean after the player has returned to the ground whether in bounds or out of bounds. I just don't think the act described in this portion of the play fit the definition of a catch, and the sideline and the contact by A has nothing to do with it.
As for #1, are you asserting that this was a muff (which I believe) and not a catch and throw (change of team possession)? If so then the result would have to be A's ball at the spot of joint possession of the catch as I stated earlier.
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