A Team can be awarded a run even when a player makes a third out only when the 3rd out is NOT a Fly out,a Line out,a Force out (such as on first base when a player has no choice but to go to that bag because he is forced by a base runner behind him) or a caught third strike.
Also the run will only count if the runner crosses home plate BEFORE The third out it made.
Ways that a third out can be made that allow the scoring of a run are in such situations as when a runner is tagged out on the basepaths (such as when he is trying to stretch a single into a double.
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It depends on whether or not the 3rd Out was a force play or not. If not a force play, the run scores if the runner crosses the plate before the other runner is tagged out. This is called a timing play.
Runs do not count after 3 outs. Runs scored before a force out is made do not count either.
As long as the out at 1st isn't the 3rd out of the inning, then yes the run would count
no it does not count, even if the plate is touched by the runner the run still does not count!