Opie Flunks Arithmetic - S5-E30
Other mistake: When Aunt Bee walks out of the grocery/meat store the door's located between the grocery's one display window and the TV display window, but the grocery store actually has two display windows with its door between the two. This was shot on the soundstage, not backlot.
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.