The Pageant - S5-E11
Continuity mistake: While Andy and Opie are sitting on the couch, Andy's on the phone talking about having to tell Aunt Bee to drop out of the play, and there are two candlesticks with candles on the server behind Andy, but in Andy's closeups they're gone.
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.