The batter isn't out on #2
Since the ball went to the screen in situation 2, the batter isn't out even if the checked swing is ruled a strike. There was a question on McGriffs about what to do if a batter did not run to 1st in this situation and I think the consensus was that the run scores no matter what since the 3rd out was made by R2.
I think is situation #1, the check swing appeal would be a valid strike out since it occured before R1 was thrown out at 2nd. This is consistent with the notion that on a checked swing on a 3-2 count, the runner and the catcher must be alert to the possibility that the ball call may be reversed and the runner can be thrown out at 2nd.
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