Whenever an offense reaches the 4th down in American Football, i.e. they didn't achieve to get a new 1st down in their past 3 attempts, they're faced with basically 2 fundamentally different options. They can either play that down as a "normal" offensive play in order to reach a new 1st down, or they can either punt the ball or attempt a field goal, depending how far they are from the end zone.
The usual strategy is to settle for a punt or field goal, unless the situation is rather urgent (i.e. you can't risk giving your opponent the offense) and/or you're confident you can make a new 1st down. What I haven't ever seen in the NFL (although, I'm only following it rather actively for about 2 seasons) is a faked field goal attempt or punt and I wonder if this is even possible. What I mean is making the play look like a field goal/punt, i.e. putting the special team and kicker on the field, and then having the kicker actually run (or pass?) the ball instead, trying to make a normal 4th down attempt.
So do the rules actually allow you to fake a field goal attempt or punt which you then transform into a normal play or does making a genuine field goal attempt require you to go through with this and not doing so would just result in a failed field goal? Is there a difference between punt and field goal in this regard? Or is there a difference between various rule sets, i.e. College and NFL? If you are technically allowed to do that to some degree in the NFL, then why isn't it done more often as a trick play?