Andy's Old Girlfriend - S7-E20
Other mistake: At the cabin, when Andy and Alice are fixing breakfast at the kitchen counter, note the cereal box's brand name is covered (presumably painted), which is still recognizable as "Post Toasties." (Andy Griffith even did commercials for this specific cereal). (00:20:00)
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.