Biscuit's post in Runners Leave Early was marked as the answer
The first part is easy, and the most common question we get. Yes, the run scores. This is a timing play, not a force play.
The second part is considerably more interesting, and involves one of my favorite things in baseball, the advantageous fourth out. Yes, they can appeal that the Runner left early and record a fourth out. If they do so, that run does NOT score. But, until they appeal that he left early at third, that run does score.
Biscuit's post in Making a call was marked as the answer
I'm not sure I understand the question... In the case of contradicting call and signal, the umpire will almost always immediately correct themselves. If they don't, the runner should stay on the bag until it becomes clear what the call actually was.
Biscuit's post in Signaling strike three was marked as the answer
There's a difference between hammering a strike, and using the out mechanic on a swinging strike three. On a swinging strike, except for on an attempted check swing, we're not actually calling anything, just acknowledging that yes, the batter swing. I saw that. Making a fist in the third strike acknowledges that yes, the batter swing, and that's the third strike, he's out. It serves the same purpose as a point, just adds a little information.