My Fair Ernest T. Bass - S4-E17
Other mistake: Barney says, "I brought the car to drive you over to Mrs. Wiley's," but later when Andy, Barney, Ernest T., and Ramona leave Mrs. Wiley's we see that they're at the house that's actually next door to Andy - though to hide this fact there's a station wagon in the driveway, and they've removed the trellis beside the door.
Other mistake: The safety goggles Barney wears is missing both of its lenses. While it's possible that Barney may wear the goggles just for the look, that possibility is negated when Barney states, "Every piece of equipment and wardrobe has a specific function: helmet - anti-crash; goggles - anti-foreign objects in eyes, that is gravel and/or bugs."
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.