catcher's interference with ball put in play.

CI is the only time the manager has a choice.

Ellsbury has a special knack for this hitting of the catchers glove... IIRC in the WS last year it was he who got called out for it though.

In OBR.

In FED, there are some other times (illegal glove, and I think one more that I can't remember right now)

Never a balk (to address a common myth).

(and I know Matt knows all this -- just adding / clarifying for others)

Balks are considered a dead ball as soon as it is called.  Funny story about the CI, last year in a fall game we had this happen.  It was the first time any of the coaches, or the umpire, had seen it.  Nobody new what to do and we just elected to take the CI.  I found out later that night from a more experienced umpire what the correct call was.

I beleive balks are a delayed dead ball in OBR - play stands as long as the batter reaches first base.  It is not up to the coach however, so if a runner is on third, balk is called, and batter hits in the infield, runner does not advance and they don't get him out at first, the coach cannot request the balk be honored to get the run. 

The balk penalty is not enforced if the batter and all runners reach the next base. All codes.

A balk is immediately dead (FED,) delayed-dead (OBR, NCAA.)

There's never a choice on a balk.  It's either enforced or ignored (or ignored for the runner and "acknowledged" for the batter) depending on what happens.

And in HS, it's always enforced.

All runners must advance for the balk to be ignored.  In your scenario, the runner from third does not advance so the balk is enforced.  And the coach never has a choice on the play or penalty.