Aunt Bee's Brief Encounter - S2-E9
Other mistake: Mr. Wheeler's license plate number is 'DC 269' which has been used in earlier episodes. Then at the end, when the locks-keys man pulls his truck into the driveway, it has plate number 'VT 772' another prop plate seen in "Barney on the Rebound" when the pretty woman drives into town.
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.