1. Yes, if you are playing OBR, once the batter leaves the dirt circle he can be called out for 'desertion' (J/R phrase). FED still allows an advance until the BR reaches DBT.
2. PUs mechanics here are incorrect - to me, pointing down at the dirt could be anything. The generally accepted mechanic is to give the 'safe' sign and verbalize 'no catch! no catch!' That lets everyone know the situation. I continue to see some MLB umpires give the 'horizontal fist' (a la Eddings) but I don't know the rationale behind that - I think the safe sign is better, because that makes it crystal that the BR is NOT out.
3. RE: abandonment, the dugout is DBT, so of course that's too late to advance. Prior to that, umpire judgment prevails - I would not be quite as generous as you in allowing their return if I were convinced they had abandoned their bases.
If you call the BR out for desertion once he leaves the circle, you avoid all this stuff with the other runners
BTW, in #1, once the BR stepped into the dugout (as you say), its too late for the coach to yell at him, he's out. It was past the 'last minute'.