Situation:

Bases loaded, 2 outs.  The 3rd strike is dropped by the catcher.  Batter does not run to 1st, but heads for the dug out.  Meanwhile, the runner from 3rd comes home, but the catcher misses the tag and R3 crosses the plate.  Meanwhile, the catcher realizes that the batter has still not run to 1st, so he goes over and tags the batter out (for the 3rd out) before he has reached the dug out (or other dead ball territory).  Does R3's run count?