I'm pretty sure as well that a free kick is not restricted to a drop-kick. If I'm not mistaken as well I think a free kick can be tried with no time left on the clock. So if time expires during a punt and a fair catch is called for, the kicker can get a shot at a field goal that is unopposed.

I believe all of this is true. I do have a question on this, though, since I have never seen it occur. Does the kicking team have to snap the ball like a regular play, or, since it is a free kick, can they tee it up or have it held at the LOS?
A
Drop Kick and a
Fair Catch Kick are two separate things. A
Fair Catch Kick works like this:A. Anytime a team makes a fair catch (for any type of kick), they have the right to attempt a field goal from the spot of the catch.
B. This applies even when there is no time left on the clock (thereby giving teams the opportunity to score at the end of a half).
C. If a team elects to make a
Fair Catch Kick, they can do it one of 4 ways:
1. Kick a normal field goal (complete with snapper, holder, LOS, etc.)
2. Line up in punt-style formation (with snapper, LOS, etc.) and
Drop Kick the ball toward the uprights
3. Kick a FG at the exact spot of the catch (without a snapper, tee or LOS; you can still use a holder, though)
4. Make a
Drop Kick at the exact spot of the catch
(had to edit because I was unfamiliar with the bold coding)