I’m not sure. Hard to see how this would happen, given the trajectory of the ball.
Incredibly unlikely, as the kicked ball is going to be traveling far faster than anyone from the kicking team could run to that point. I have no idea if it’d be legal or not.
Not legal at all. It has to be kicked.
Not legal, in either case (punted or thrown). After the initial kick, the ball can’t be legally kicked a second time (at least, not in American football). And, as noted above, a thrown ball can’t be a legal field goal.
See above for why this is incredibly unlikely. I have no idea what the actual ruling would be, but I have a very hard time believing that the kicking team could score any points from it.
I’m not sure about the NFL, but in the NCAA, it wouldn’t count, according to rule 8-4-1a: “A field goal shall be scored for the kicking team if a drop kick or place kick passes over the crossbar between the uprights of the receiving team’s goal before it touches a player of the kicking team or the ground.”