If there are less than two outs, yes. If there are two outs, this is a timing play. If the runner crosses home plate before the batter is thrown out at second base, the run counts. If the batter is thrown out at second base before the runner crosses home plate, the run does not count.
The only way that run would count is if it were not a continuous double play... For example, if the batter grounds into a routine double play, the run does not score... but if the batter were to fly out and another runner tries to tag up, and advance from 1st the second, or from 2nd to 3rd, and is thrown out in the process, as long as the runner for 3rd crosses home plate before the out is recorded, it counts. To simplify the answer a bit... If the double play occurs with both outs being recorded on force outs, no run scores. If there are multiple outs recorded, but are not force outs, the run scores as long as the runner crosses before the out is recorded.
Yes. If a runner crosses home plate before the third out is made (unless it's a force out), the run counts. For example, say the Yankees have runner on third with two outs. The batter hits a ball to the gap in right center. The runner scores, but the batter is thrown out at third trying to stretch a double into a triple. Since the runner on third crossed home plate before the batter was out at third, the run counts. On a force play (at any base) the runner would not be allowed to score even if he crosses the plate before the force is made.
No, because the 2 part means the runner on third was thrown out
nope, as long as you get the out no runs count.
If the runner who started on 3rd base is the one thrown out at home, no it is not a hit. It is ruled a fielder's choice
A run counts if the runner crossed the plate legally and before the third out. If the runner crosses the plate even a half-second before the third out is tagged on the basepaths, then the run counts. If the out is made a half-second before the runner crosses the plate, then the run does not count. One exception is that with two outs the runner cannot score until all forced runners have reached their bases safely. That is, if the batter-runner is thrown out at first even after the runner crossed the plate, then the run does not count. Same thing if a runner is thrown out at second or third base (on a force-play only).
Yes, as long as the batter thrown out at first is not the third out of the inning.
no, but the base runner is out if he is struck by a BATTED ball (but he isn't out if he touches a ball thrown by a fielder)
The runner on second base can be thrown out provided that first base is empty
Presumption - you mean "Does the run count if the runner crosses home before the THIRD out is committed?" It depends on the nature of the third out. If the out is a force out (e.g. fly ball out, out on an appeal, or force out at a base the runner must advance) then the run does not count. If the third out of an inning is a non-force play where the runner does not have to advance (e.g. a runner stretching a single into a double is thrown out at 2nd), then the run counts if the scoring runner reaches the plate prior to the out being made.