First post, bear with me please. Yesterday doing an 11 y.o. Pony Tournament--2 out, R2 hit by batted ball in the hole before anyone touched it, clearly out. Inning over.
Then I realised, what if there had been one out? What happens to the batter? Does he get first without liability and R1 to second? Thanks-I learn a lot here.

You got it right. Ball is immediately dead, award batter 1B and any runners forced to advance will also advance.

I think you are right here. Where it may get confusing is if you have batter interference with R3 attempting to score at home. With less than 2 outs, R3 is called out and the batter remains at bat. With 2 outs the batter is out and the person who follows him will be the next batter in the next inning.

You're right but isn't it nice to realize you have to find an answer before it blows up on you.