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Situation: 9 year olds - Pony Mustang level. Runner on third. Batter bunts and catcher comes forward to field the ball and throw to first. First baseman sees the runner on 3rd start for home on the throw to first, so he guns the ball home, but the catcher wasn't at the plate. Catcher was still in infield. Runner scores and batter advances to 2B.

Mental error for the catcher not covering home, but the coaches want to charge an error to the first baseman for essentially throwing (clean throw had the catcher been there) the ball away. I viewed that as a mental error as well.

What do you think?

It depends. Would the runner from third have scored even if the catcher was behind the plate? If the runner would have scored anyway, then there is no error. If the runner would have scored, then I would charge an error to the first baseman. It doesn't matter if someone should have been at the base, it was the first baseman's physical misplay of the ball that allowed the runner to advance home (and incidentally the batter to advance to second).

Firstly, you can not error any mental error, there must be some form of misplay.
I am assuming that the throw from the 1st basemen was wild because he threw the ball at home plate, not to the catcher. If, in the scorer's judgement, an errorless play would have seen the runner from third definitely out at home then it is an error on the 1st baseman. Given that this is a tag play the scorer would have to be certain that the catcher could have caught the throw and applied the tag to make the out.

In this case there is an error because it is how the batter advanced to 2nd on the wild throw. The question is if the runner scores on the advance or the wild throw. The batter reaches 1st on a fielders choice.

This, like so many other scoring questions, is one of those HTBT situations. OBR 10.12(a)(8) states:

The official scorer shall charge an error against any fielder...whose failure to stop, or try to stop, an accurately thrown ball permits a runner to advance, so long as there was occasion for the throw. If such throw was made to second base, the official scorer shall determine whether it was the duty of the second baseman or the shortstop to stop the ball and shall charge an error to the negligent fielder.

Rule 10.12(a)(8) Comment: If, in the official scorer's judgment, there was no occasion for the throw, the official scorer shall charge an error to the fielder who threw the ball.

So, according to 10.12(a)(8), if in the scorer's judgment, (i) there was occasion for F3's throw home, (ii) F3's throw was accurate, and (iii) F2's failure to stop F3's throw permitted one or more runners to advance, then charge F2 with an error. If there was no occasion for F3's throw, then charge F3 with an error. In either case, 10.12(a)(8) addresses how to deal with the failure to stop an "accurately thrown" ball.

In most situations, an "accurate" throw is one that is made to the fielder, not the base (one exception being a throw to second on a stolen base attempt when the throw is made to the base while either F4 or F6 move over to the bag to make the play, which is the basis for the second sentence of the rule). So, in the OP's case, 10.12(a)(8) doesn't apply, as the throw was not "accurate."

OBR 10.12(a)(1) states in part:

The official scorer shall charge an error against any fielder...whose misplay (fumble, muff or wild throw) prolongs the time at bat of a batter, prolongs the presence on the bases of a runner or permits a runner to advance one or more bases...

So, in the OP's case, charge F3 with an error if F3's wild throw either (i) permitted R3 to score (i.e., an "accurate" throw to F2 would have prevented R3 from scoring) or (ii) allowed BR to advance to 2B.