Given the USCF rules for stalemate and Checkmate (immediately ending the game), are directors allowed, and/or are they required to declare a game over when noticing a mated position?
If the above is allowed and/or required, do the rules require that a director correct a position that was (check/stale)mate when the players continue moving, i.e., players are unaware?
I don’t have it in front of me, so I’m paraphrasing, but rule 18G2 says that directors are allowed to declare checkmate or stalemate even if the players don’t notice. It says “allowed” rather than “required”, so there is some TD discretion available. I believe this is the only wording on the subject other than that checkmate and stalemate immediately end the game.
It’s a bit Zen, really. 
If a legal move has produced either checkmate or stalemate, the game is immediately over. Checkmate or stalemate does not have to be “claimed.” A TD who observes a checkmate or stalemate position on the board and records the result appropriately is not intervening in a game in progress, simply because the game is over.
But … the TD had better be absolutely sure he’s right (that the position is checkmate or stalemate), and there’s the rub. (Note the currently active thread about a player who “claims” checkmate and the opponent appears to agree before a spectator interferes.) Personal anecdote: About eight years ago, I was directing a G/60 event. I was watching the last game in a section. Both players were in extreme time pressure, and pieces were flying. To this day, I’m fairly sure I saw Black make a knight move that checkmated White’s king. Neither player seems to have noticed that the King was in check, and after maybe four moves Black moved the knight away. But the players were moving quickly enough that I could not be absolutely certain the position I saw was a checkmate. Obviously, I did not intervene, but I still remember that as a TD lesson.
I looked it up in the Kindle version (6th edition).
18G is about how directors should not permanently adjudicate games except under emergencies.
One of my students told me “coach you have to forgive my blunder, for quite often
I do not think all the time” 
Rob Jones
By any chance was that Winnie the Pooh?
Bill Smythe