Does this run count?

Yes, assuming R3 reaches home before first base is tagged.

Per rule 5.08(a) EXCEPTION: a run does NOT score if the third out is made, 1) on a force out, 2) on the batter-runner before he reaches first base, 3) on a preceding runner.

In this situation, the putout of the runner at first is NOT a force play. It is a live ball appeal for failure to tag up, which is never a force play because the batter is out on the fly ball.

There is a common misconception that a force out occurs any time an out can be made by tagging the base. That is not the case.

A force situation occurs when a runner loses his rights to a base when the batter becomes a runner. But, as soon as the batter-runner is put out, all force situations are removed. So in this situation, when the batter hits the ball, R1 is in a force situation, but when the fly ball is caught, the force is off.

The reason that R1 can be put out at first is because a runner is subject to being put out, on appeal, when the runner has not touched his original base at or after the time a fly ball is first touched. (The runner has to "tag up.") The runner is put out in this live ball appeal by either tagging the runner or his original base.

Yes, the run counts.

Technically its a judgement call for the home plate umpire to determine whether or not the runner from 3rd base touched home before the 3rd out was made, but in this case the runner from 3rd would almost certainly score before the out.

This situation is a common question in rule related quizes because there's a common misconception that the 3rd out would be a force, which would nullify the run.

However, rule 2.0 definitions clearly defines a force play as one on which a runner is forced to advance because the batter becomes a runner. When a runner is returning to his original base after a caught fly ball he is not advancing. Therefore this is not a force play. Its actually considered an appeal play.

If the runner from 3B crosses the plate before the out is recorded at 1B, then yes the run counts as this play is not a force play, it is an appeal play but not a force out.

The run counts.

As long as the runner from third touches the plate before the third out is recorded, yes.

No. The third out is made on the play. Not on the continuation of the play.