scoring ahead of the third out

Question
slow pitch asa  9-12    runner on 1st and 2nd and third   fly ball to short left caughtfor 2nd out-runner on2nd tagged trying to return for the third out  runner on 3rd scores before the tag does the run count?            SAME GAME later inning  2outs runner on 3rd  batter hits ground ball to 2nd base  fumbled around but gets the out at 1st base run clearly scored ahead of the out does the run count?

Answer
Thanks for the question!

The answer to the first part of your question is yes the run would count.  However, say that the runner at third didn't tag up properly and went home and scored anyway and after tagging the runner trying to get back to second to record the 3rd out, the defensive team appeals the runner left early.  This is called a 4th out appeal which results in the run not counting.

The answer to your second part is no the run would not count.  Runs are not counted when a player is out as a result of being forced by a batter becoming a batter-runner(in this case the batter is forced due to becoming a batter-runner).  Even if instead of getting the force out, they tagged that runner at 1st for the third out, the run would not count.

I hope this answers your question, if not please reply!