Under National Federation Rules:
Where is the ball put into play if a free kick after an awarded fair catch goes out of bounds untouched by R? At the spot where it goes out of bounds as in the case of a scrimmage kick? Or does R have the same options under the free kick rules-(35 yds from previous spot, spot of out of bounds, 5yds from previous spot and (re-kick) replay down)?

Other than 25 (not 35) yds. from previous, why not?

See rule 6-1-8. A free-kick is a free-kick, only difference is that the kind you ask about can score a field goal.

I did get that crossed up...25 yds...thanks

The confusing part to me is that an attempted field goal is a scrimmage kick by definition...and after an awarded fair catch the kick becomes a free kick...so if it can score shouldn't it be treated as if it were a scrimmage kick after the attempt is made?

Nope, the free kick after a faircatch is the exact same thing as a KICKOFF, only difference is that it can score.

No, it's not.

8-4-1a
The field-goal attempt shall be a place kick or a drop kick from scrimmage, OR from a free kick following a fair catch, or an awarded fair catch.

OR means it can be either.

Had one Friday night. See the Fed board for the details.

NF RULES: Is a team that makes a fair catch or is awarded a fair catch permitted to free kick with a punt? I know that they cannot score unless the free kick is a placekick or dropkick, but is it permissable? Rule reference, please.

NF only: NO, the only a way to score 3 points from a free kick is via drop kick or place kick.
ref 8-4-1-a, but the definitions are the specific rules you want.. 2-23-6 and 2-23-7.
The PUNT is not one mentioned.

After a fair catch or an awarded fair catch, R-A can do anything they want. They can run, pass, punt, place kick, drop kick, take a knee, run into their own EZ and give B a Safety.

If they run a pass play, and DPI is called; they get to do all over again, 15 yards closer.