It's ruled "caught stealing," and scored 2-6 if the catcher throws to the shortstop, 2-4 if the catcher throws to the second baseman, etc.
Technically, yes, it is illegal to advance the runner. The official penalty is called "Helping the Runner." It is rarely enforced, however.
If a runner is touched by a fair ball in fair territory before the ball has touched or passed an infielder, the runner is out. The ball is dead and no runner may score. And no runner can advance, except runners forced to advance. [Rule 7.08(f)]
The courtesy runner is credited with scoring the run.
No, he can advance by stealing the base, or advance on a wild pitch, passed ball, catcher interference or a pitcher's balk.
That is a term generally used when a base runner is stealing a base. The catcher will not attempt to make a play on the runner (throw the ball to the base that the runner is attempting to steal). Sometimes, the official scorer will rule this 'defensive indifference' and not award the runner with a stolen base. You might see this in the 9th inning of a game where the team batting is down by several runs and the defensive team is more concerned with getting the batter out than attempting to throw out a base runner attempting to steal since that run will not affect the outcome of the game.
In baseball "advance the runner" is a term used to say get a runner on base to move forward to another base.
No because the batter had nothing to do with the runner scoring.
runner in scoring position
According to MLB rule 7.08(f) "A runner is out when ... He is touched by a fair ball in fair territory before the ball has touched or passed an infielder. The ball is dead and no runner may score, nor runners advance, except runners forced to advance." Since the ball is dead the instant it touches the runner, where it goes afterwards is irrelevant.
The player who passed another on the base paths is out, the runner that was passed can continue to run to the next base.
The only other way I can think of that a runner would score from a fly-out without tagging up would be when the fielder catches the fly ball and then throws it into the infield and the infielder doesn't catch the ball properly, allowing the runner to advance home on an error.