Sure, as long as he keeps it in his hand while fielding the ball.
Rule 7.05 states:
Each runner including the batter-runner may, without liability to be put out, advance-
(b) Three bases, if a fielder deliberately touches a fair ball with his cap, mask or any part of his uniform detached from its proper place on his person
(d) Two bases, if a fielder deliberately touches a thrown ball with his cap, mask or any part of his uniform detached from its proper place on his person
In other words, if a fielder intentionally touches a batted ball with his cap while it is in his hand, all runners, including the batter, advance three bases (ie, every runner on base scores) from where they were when the infraction occurs.
hitter is awarded 3 bases
He moves on!
Yes, it would be fair as long as it is fielded before first or third base or comes in contact with the base.
If a runner in fair territory is struck by a batted ball prior to the ball having been fielded, the runner is out.
If the ball is fielded in the end zone and the player does not wish to attempt a return, usually the player will "take a knee" to kill the play. The player could also simply run out of bounds in the end zone. Aside from that, the officials typically would not blow the play dead unless it was clear that all players on the field had essentially given up on the play and no effort was being made to advance the ball. But this would be entirely at the official's discretion.
players, match officials and supporters would float in mid air. The ball when kicked would fly off in to infinity unless a player could catch it in flight. Basically the game just would not happen
It explodes
Someone would be screaming in pain.
The kinetic energy would transfer Ping Pong ball!
The ball would roll downhill.
Generally speaking, it would be a hit unless the fielder clearly had enough time to throw out the runner and did some other action instead (such as an attempt to tag a base runner, but not throw the ball).
You would fall
You would get Lugia.