Something new to me...

[SCHSref]

R1 fair catches a scrimmage kick at K's 40. He then calls TO. After the TO, he informs white hat he wishes to put ball in play via snap. A1 passes to A2, but B1 interferes and pass is incomplete. After penalty administration, they decide they now want to free kick from K's 25. Apparently, this is permissible. I never knew...anyone ever see this?

[Ralph Damren]

Electing to free kick after a fair catch is as rare as Halley's Comet. If B fouls on the first scrimmage down after a fair catch, it's sorta' like said comet landing in your back yard. But yes, they still have that option. This could also occur in reverse :
(1) Fair catch at K's 35-free kick option chosen;
(2) Kick is good while K player punches R player;
(3) bad kid gone and ball moved back to 50, out of kicking range;
(4) K can turn to A and take scrimmage downs.

While this is on my Bucket-list, I may kick the bucket before 

[ttown44]

Key being "replay the down" which includes all of its options, including the placement of the ball between the hashes.

[CalhounLJ]

Electing to free kick after a fair catch is as rare as Halley's Comet. If B fouls on the first scrimmage down after a fair catch, it's sorta' like said comet landing in your back yard. But yes, they still have that option. This could also occur in reverse :
(1) Fair catch at K's 35-free kick option chosen;
(2) Kick is good while K player punches R player;
(3) bad kid gone and ball moved back to 50, out of kicking range;
(4) K can turn to A and take scrimmage downs.

While this is on my Bucket-list, I may kick the bucket before 

Had my first one this year.. after 30+ years of officiating. R fair caught the ball at the K 46 with :01 left in the half. Kick was right down the middle but short by 2 yards. Even though the ball was never touched, the clock operator ran off the :01. Nobody complained, so we went to halftime.

[ncwingman]

Had my first one this year.. after 30+ years of officiating. R fair caught the ball at the K 46 with :01 left in the half. Kick was right down the middle but short by 2 yards. Even though the ball was never touched, the clock operator ran off the :01. Nobody complained, so we went to halftime.

As you noted, the game clock does not have to run on a free kick play (it would on a scrimmage play, since it would start with the snap). As a deviation from the original question, how many consecutive free kicks do you think you could have such that no time runs off the clock?

In theory, there could be an infinite chain -- if the fair catch kick is good, the ensuing kick off could end with a fair catch or awarded fair catch without the ball being touched by R (KCI, untouched by R), that team fair catch kicks, and the whole process just repeats forever.

[SCHSref]

The fair catch after a free kick is not new to me...the running of a play and then free kicking IS. AS someone else pointed out...the "replay the down" is the caveat

[NVFOA_Ump]

As you noted, the game clock does not have to run on a free kick play. ....

Don't want to be nit-picky here but shouldn't we say that the game clock does not start on a free kick until the ball is legally touched in the field of play?

[Ralph Damren]

Don't want to be nit-picky here but shouldn't we say that the game clock does not start on a free kick until the ball is legally touched in the field of play?

We should, lt's a free kick. Differing from the free kick after a score, we would set the chains to mark the neutal zone and place R &  sNiCkErSU to make the call.     

[sczeebra]

SCHSref, remember this twist from our rules interpreter Bruce a few years back where the offended team(penalty inside two minutes of the half or 4th) decided to hold off starting the clock until the snap. There was a play with an inadvertent whistle with a replay of the down and that offended team changes their mind and wants to have the clock start on the ready which was ruled acceptable. Merry Christmas!!!

[GA Umpire]

While this is on my Bucket-list, I may kick the bucket before 

Ralph:
I have had one in my long officiating life. 
Have had coaches ask in pregame meeting if it was still a rule and state they always have wanted to try it bur have not had the chance.