Mayberry Goes Bankrupt - S2-E4
Continuity mistake: In establishing scenes throughout the first part of the episode, Andy is seen driving up to Meyers' home which seems to be located far from neighbors and on a dirt road. In the "renovation" scene and those following, Frank's home is located on a paved road with other homes nearby in the shot. Clearly a different set entirely.
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.