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For an improper batter to be called out, they must complete their plate appearance.
With a runner on first, I'd interpret the wild-pitch question differently than Paronto did, because the runner was forced to second, and furthermore the defensive team was denied the chance to put him out. Under no circumstance can an improper batter becoming a runner give the batting team an advantage.
With a runner on second and first base open, the advance stands because the runner's advancement was not affected by the improper batter becoming a runner. However, if the runner's advancement to third is a result of a play being made on the improper batter-runner, he will be forced to return to second.
If a runner on second (with first base open) attempts to steal third on ball four to an improper batter and is thrown out, the out stands and the proper batter is also out.
If a runner on first overslides second base on ball four to an improper batter and is thrown out, the out stands and the proper batter is also out.
In the strikeout/wild pitch scenario, the runners were forced to advance and would return to their bases at time of pitch, but it's irrelevant because the proper batter is the third out. Since the putout goes to the catcher it can be assumed that the proper batter is considered to be put out before he reaches first base.
In the strikeout/wild pitch scenario with less than two outs, the batter is out, so the runners are not forced and do advance because the pitch being strike three does not affect their advancement.
If there is a force play at second with less than two outs, the out at second stands and the proper batter is also out, because the rule for batting out of turn contains no provisions for returning runners who have been put out to their time-of-pitch base.
If there is a force play at second with two outs, or a double play that does not involve the batter-runner with one out, the only person ruled out is the proper batter because otherwise four outs would be recorded in the inning.
All scenarios assume that there is an improper batter and that an appeal is made. I am not an umpire but I know the rule and know how to enforce it.
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