Bounced Pitch & Dropped Third Strike

Question
Less than two outs, first base open. Runner on third. Pitcher pitches the ball, it bounces before crossing home plate. The batter does not swing. The Umpire incorrectly call "strike three". The batter runs to first after the coach screams "dropped third strike, run". The catcher throws to first, overthrowing. The runner from Third scores. The batter/runner advances to second.

Questions. Is this a "uncaught third strike". I understand that the pitch SHOULD have been called a ball by the umpire, but since he called it a strike, and since it bounced before being caught (so it was not a clean catch) is the runner entitled to 1st base if not thrown or tagged out?  

IF it is not a dropped third strike, then does the run still score, since in that case the run wouls simply be a stolen base?

Answer
Rick, thank you for your question.

I'll bet this one got pretty exciting.

The coach was right to tell his batter to run, since once the pitch hit the ground and first base is unoccupied, the umpire has called the pitch a strike, he has the right to try to reach first base.

The rest of it plays itself out, the run scores on the catcher's throwing error.  The batter is safe at first on the catcher's fielding error, then safe at second on the throwing error.

When the dust cleared, the coach in the other dugout should have come out and asked the home plate umpire to get some help from the base umpire, since the pitch was in the dirt, the batter did not swing.

The base umpire is in a position to see it all; but he cannot interject on the plate umpire's call, unless the plate umpire asks him what he saw.

The plate umpire is under no obligation to ask for his opinion, just because a coach requests it; but most will do so.

I would guess by the time this played itself out, there was a lot of controversy coming from dugouts, bleachers and parking lots.

The base umpire should have corrected the call to be a ball, since the pitch hit the ground.  Depending on the number of balls in the count already at the time, the batter either walks or continues to hit.  

The runner from third goes back to third, and calm is once again restored to the game, although the discussions of that initial ruling will go on for years I'm sure.

From your question, it sounds like no one asked the plate ump to ask for help, or if they did, he refused.  Once it goes there, coaches are out of options, time to live with the judgement call and move forward.  Everything would remain as it played out, run scored, less than two outs and a runner on second.

Yours in baseball,

Rick