Dropped 3rd Strike, can R1 advance to 3rd?
 

NFHS Rules. Runner on 1st, 1 out. Dropped 3rd strike on the batter. (He's out and cannot be safe at 1st) R1 takes off from 1st because the ball got by the catcher. B/R runs to 1st because everyone is yelling "RUN". The catcher gathers the ball and throws to 1st to get the B/R, on the throw R1 goes to 3rd.

To clean this up, the umpires call the B/R out and allow R1 to remain at 3rd.

Can R1 stay at 3rd or should he be returned to 2nd because the B/R caused a throw to first after already being out?

Runner stays at 3rd.
The defense is supposed to know the situation.
Had they thrown him out at 3rd the out would have stood.
The B/R is still out.
The B/R is not guilty of interference just because he runs to first when he is not eligible to advance on an uncaught 3rd strike.

Same ruling in all codes.

PU should be pointing to B/R and loudly and emphatically yelling "He's out! He's out!"

True, but half the time nobody listens, or hears.

Even so, it's one less thing for the defensive coach to argue.

Proper mechanic is for the umpire to shout, "Batter is out" when he starts his run to first base.
If he continues the PU says, "Batter is still out!!"
If the action continue just sit back and let the circus begin.
Any thing that happens--runners advancing, scoring, being thrown out, etc will stand.
When the smoke clears point to the batter ( if he is on base somewhere) and say,
"He's out"!!