The Darlings Are Coming - S3-E25
Continuity mistake: When Briscoe gets into his truck, his hat is wet from dipping it (5x) in a horse trough. The camera cuts away to Andy and immediately back to Briscoe, whose hat is now dry. Also, "she's always hood for 11 hat fulls," yet Briscoe only puts 5 hats of water in his engine.
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.