Balk Rule Explanation
 

NCAA game, runner on first, one out, pitcher from the stretch does not come to a stop and pitches the ball. Umpire calls balk. Batter hits a single between right fielder and center fielder. Here are my questions:
1. Legal play, play stands and continue game from that point. One out, runners on first and second.
2. Runner on first tries to reach third on the single and is thrown out by right fielder. Play stand or can offensive team now elect to have balk enforced?
3. Once each runner on base has safely advanced one base, is balk ignored, or does offensive team have a choice to take balk at anytime before the next pitch?

runner gets protected to the base he would've achieved on the balk...runner chooses to advance at his own risk beyond what the award would be.

The offense never has a choice on a balk. Once each runner advances a base, the balk is ignored. If R1 is thrown out at third, as in your play #2, the out stands.

If you are guessing ... please post that you are guessing. Someone is going to come on here, read a post, see your incorrect answer stated with conviction, and then go misquote the rulebook somewhere. (PS your guess is not just wrong at NCAA, but at many other rulesets as well).

edited to stand correct in rulesets except for FED in which case the balk kills the play...

That wasn't the only problem with your post.

"This particular sitch...the offense should take the base hit"

Here you give the impression that the offense has a choice...they don't.

I quote my own edited post.

Bob Jenkins is correct. Rule references to support: pg 105 of 2007 NCAA Baseball Rules

NCAA 9-3-j: defines pitcher failing to come to stop from stretch position before pitching ball it is a BALK.

PENALTY for a.through m.- Balk. The ball becomes dead and each runner must advance one base.

(1) If the balk immediately is followed by a pitch that permits a batter and each runner to advance a minimum of one base the Balk is IGNORED and the ball remains dead.

So no matter what scenario you can come up with the play stands and the offensive team will never have a choice because the balk is IGNORED.