Three Wishes for Opie - S5-E14
Revealing mistake: Barney, convinced he can foretell the future, is in the darkened jailhouse, seated between Goober and Floyd at a table. Goober strikes a match and lights a lamp which is directly in front of Barney. The immediate illumination obviously comes from a studio light because the men's shadows on the wall are to the far left of the men, instead of directly behind them.
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.