As others have said, forget the term catcher's balk. Focus on the catcher's interference, the penalty of which is a balk on the pitcher and the batter gets 1B. Yes, doesn't only apply with a R3, it applies with all runner combinations.
Assuming, in the mind of the umpire, it was an unmistakable appeal (most likely would be), the runner would be out. This is a timing play. If the runner scores before the runner is tagged, the run would count.
As with any written rule you have to look at interpretations that go with it. Does this quote make you feel any better?
"If, in the judgment of the umpire(s), this behavior is occurring, both the player and the manager may be ejected from the game. It is the discretion of the umpire to decide if the action is blatant and consistent to the point where ejection is warranted."
https://www.littleleague.org/university/articles/hey-blue-protesting-unsportsmanlike-conduct/
Umps screwed up. There is no right way to fix this. You do the best you can. Defense made a decision based on an erroneous umpire call (out at home). Give a little to each side. I would have left R2 at 3B.
Not golf. As long as they don't increase intensity right when the pitcher releases the ball. Don't be ridiculous and keep the chant going through the whole pitch.
The only "judgement" required here is that the runner did not intentionally interfere with the ball, which almost never happens. Otherwise what happens happens when a runner is hit by a deflected ball.