Andy and Opie - Bachelors - S3-E4
Revealing mistake: After Andy and Opie choose between chicken and pot roast using a nickle, when Opie gives the nickle back to Andy the window over the sink is missing - the fake window wasn't put up on the set wall for this shot, but when the food burns the window and curtains are up there as usual.
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.