I finally found the right answer to this question. I stumbled across it in an “Ask the Referee” online column on http://www.ChicagoSports.com
In the situation described above, former NFL official Jerry Markbreit says that the illegal touching would be figured from the point that the player last touched the field. That would be the 1-yard line.
So, in the situation described above, the receiving team would get the ball at the 1.
I’m a little puzzled. DRY said that anytime a fair catch is made, the team can opt for a free kick (field goal attempt). But Sofa King said that it can do that only in the last two minutes of a half.
You can ask for a free kick field goal try after any fair catch at any time. The rule only exists in the NFL and high school. It’s not in the college rule book.
I’ve only seen one attempted and it came right before the half ended. I believe that if you call a fair catch as time expires, you still get a chance to attempt the kick.
I don’t know what the rule states, but I guarantee that no coach would ever opt to attempt a field goal (as opposed to run normal offense) if time wasn’t running out. The team must have great field position if a field goal is possible, and with plenty of time they’d surely attempt to drive for a touchdown.
It almost happened again at the end of the first half of the Packer/Buc game. Dungy appeared all set to have his team make the fair catch and then try the kick, but Sherman was on to him and sent Favre in to kill the clock instead.
Here’s a quote I posted in another thread a month or two back when we were talking about the kicking game:
"IIRC, the last attempted pro game dropkick was in 1949 on a botched field goal snap. The kicker ended up with the ball and tried to dropkick it through, but I think was unsuccessful. (He may have made it, I forget.)
In recent years, QB Jim McMahon was a devotee of the dropkick and would practice it for hours. Apparently he got quite good at it too – no small trick in this day and age of pointy footballs. Whenever his team would get into range he would nag his coach to let him attempt it, but they never did."