If gravity were magically turned off, would the moon move away from the earth? No. The moon's motion away from the earth is caused by the fact that tidal friction, which is caused by gravity, slows the earth's rotation. This represents a loss of angular momentum in the earth-moon system. To conserve angular momentum, the moon must move into an orbit with greater angular momentum. This is an orbit that is farther from the earth. The angular momentum does not cause the moon to retreat from the earth. The CHANGE in the angular momentum is what causes the moon to recede. The change in the angular momentum is caused, as you have already conceded, by gravity.
It most certainly would. Its momentum would carry it in a straight line in whatever direction it happened to be moving at the time of shutoff. That would carry it farther and farther from the earth.
(You did NOT turn off momentum!)