Continuity mistake: When Barney disguises himself as an old woman, in the very first shot he's wearing a dress with lace and tiny flowers, but in all the following shots it's an entirely different dress with large groups of flowers. (00:17:40)
Visible crew/equipment: When Barney and Andy are waiting for Virgil to arrive, just as the bus pulls up and stops, the reflection of the boom pole/mic is visible on the windshield. (00:03:50)
Aunt Bee and the Lecturer - S8-E10
Visible crew/equipment: When Andy parks the squad car in front of the house he asks Hubert about the picnic, and the reflection of the moving boom mic is visible on the windshield. (00:12:35)
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.