At umpire school, the "out" mechanic, is also the "strike" mechanic. This is the very basic mechanic for these things. Just like the "safe" mechanic is used for other things too, like no catch or no tag, or "ball is still in play." Fans may be confused by the "out" mechanic being used for a strike...but players won't. Players have seen this from rookie ball all the way up. Some umpires point, some umpires use the hammer (what you call the out mechanic). This season I used the hammer. Never did I have any problems like this. The players weren't paying attention to me. The players weren't paying attenion to Eddings. The catcher didn't throw the ball to the pitcher because of anything Eddings did.
If I asked the hypothetical question you asked me to a professional umpire...or asked it at umpire school, the reply would be, "Was it a day or night game?" In other words...did it happen...lol. If a fight happens, and it's an uncought 3rd strike, then i'm going to call time and not give him first. He gets the right to run until he gets to the dugout, but using "common sense and fair play" 9.01c in the rulebook, I'd say that by going to fight, he gave up his right to go to first. You can use common sense and fair play in anything that happens that isn't addressed in the rules. A batter going towards the dugout after a dropped 3rd strike is covered, saying that he can run til he gets to the dugout, but it doesn't cover if he goes to fight. I think it's common sense that he gives up the right if he wants to go fight instead.
What do you think is fair? Sometimes an umpire just has to go with what he thinks is fair, when it's something that's not in the rules. But what happend today was in the rules. Uncaught 3rd strike. Batter starts for dugout, then tries to run and is safe.
It's nice to know what i'm talking about when it comes to baseball. Nobody else has a clue. We need an official from football on this site. Somebody who is experienced, as having done college football and stuff. That would be useful.