Three Wishes for Opie - S5-E14
Other mistake: When Andy asks Floyd and Judd about who he's supposedly marrying it cuts to Helen walking down the street, and the grocery store has only one display window, when it should have two windows, and there's a brick wall adjacent to the window which shouldn't be there. (Shot at the soundstage not backlot).
Three Wishes for Opie - S5-E14
Other mistake: When Barney uses the gypsy lamp and cards in the back room, in the shots with Goober and then with Opie while they're standing in the doorway, the wall opposite them has a barred window, but in earlier episodes that wall has the door that leads outside to the side of the building.
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.