Seahawks/Redskins Fair Catch Question

[Suudy]

At the end of the first half in the Seahawks/Redskins game, Seattle called timeouts, ending the Redskins drive on 4th and some distance with about 3 seconds on the clock (the clock was stopped).

The announcers (Tom and Cris?) mentioned something about "that old rule that on a fair catch, you get one play, even if the clock runs out." Now, I know that announcers are often the worst sources of information, but it got me wondering.

If a period (or half) ends with a fair catch, does the receiving team get one untimed down in the NFL? NCAA? NFHS? I don't have my books (Fed) with me so I can't look that one up.

Also, I was watching the game at a sports bar when Seattle misjudged the kickoff and the Redskins recovered the kick deep. Everyone (in Seattle, where I was watching it) clapped their hands to their faces and weeped, assuming the TD. Even the announcers were unclear at first. One of those situations where being an official alleviated the fears of my buddies, and made the situation when everyone looked totally confused about it not being a TD even more funny. The expressions on their face were priceless. That is a situation so rare, that it is great to see the officials get it right.

(A similar situation happed a year or two ago in a Seahawk game, but on a punt that was muffed. Seattle recovered and started to advance, but the officials nailed it declaring the ball dead at the point of recovery. The fans were irate.)