Balk Rule Explanation

[VaCoach]

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?