As others have established, the current rule governing punts, NFL and college, is very similar to that governing passes: punts may only be executed on scrimmage plays, before a change of possession, behind the line of scrimmage.
This wasn’t always a case, and there was a time when both “return kicks” and punts from beyond the line of scrimmage were legal. I’ve never seen a “return kick”, but I’ve read that they were not unknown in early Twentieth Century football. The rules and the shape of the ball in that era made passing much more difficult, and football was more of a grind-it-out field position game. If the other team was over-pressing on punt or kickoff coverage, you could gain field position by kicking the ball back to them and hoping that it would travel over everybody’s head and bounce and bounce.
I have seen a punt from beyond the line of scrimmage, at a time when such a play was legal. It was executed on fourth down, by Danny White, in a Monday Night Football game in 1981. He fumbled a poor snap in punt formation and had to scramble. When an NFL punter has to punt on the run, his team often incurs a penalty because the gunners go downfield before the kick. However, the penalty didn’t apply when the punter crossed the line of scrimmage. White deliberately crossed the line of scrimmage before punting and thus avoided the penalty. (The refs threw a flag but then picked it up, and explained why.) At some time afterward the rule was changed, and beyond-the-line punts are now illegal.
The non-fourth-down quick kick, of course, is still legal, and if one watches enough NFL or college football, one will see perhaps one such kick per season.
Don’t know if there’s any rules against it, but I would think it’s a lot harder to drop-kick with the accuracy needed to score a goal, compared to place-kicking. With a punt, the goal is usually distance, with some general aiming (e.g. - coffin corner kicks), but not the same precision needed for a goal.
To add to what Piper wrote – no, a punt can’t be used to kick a field goal as the ball doesn’t touch the ground. A drop-kick can be used (essentially executed like a punt, except that the ball drops all the way to the ground before shoe contacts ball).
Joe Theisman did it at least once as well (for 1 yard…it went almost perfectly sideways), and Danny White, of course, was both the Cowboys’ QB and punter…probably the last player ever to simultaneously hold both positions.
The rules say that a field goal or kick for extra point my be scored only from a legal scrimmage kick by place-kick or drop-kick. Place-kick is defined elsewhere in the rules as “a kick by a player of the team in possession while the ball is controlled on the ground by a teammate.”* Drop-kick is defined elsewhere in the rules as “kick by a player who drops the ball and kicks it as it touches the ground.”*
Tapping the ball on the ground, then punting it, is just a punt.
*NCAA wording. My NFL rulebook is at work, but the wording is substantially similar.
ISTR that Cunningham did so in a playoff game, and that it set the record for longest punt, bouncing for 20 or more yards, and was downed inside the 10.