You can advance your own fumble if the play is still alive, you pick up the fumble in a legal position (off your knees and elbows and not being touched by the opponent) and whistle doesnt sound.
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The receiving team can advance the ball once they gain possession. The kicking team cannot advance the ball unless it was first in possession of the receiving team.
The rules for this scenario would differ based on which league was being discussed.
Yes.
There was a rule in College Football where the defense could not advance the ball after recovering a fumble but it was abolished in 1992.