Does the run count if the the runner crosses home before the out is committed?

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.