The Church Benefactors - S8-E20
Visible crew/equipment: At the start of the finance committee meeting, when Clara walks into the office modelling the choir robe, the shadow of the boom mic is visible on the door, in the top right corner of the screen. White tape marks are also visible on the floor, as Clara twirls in the choir robe. (00:08:15)
The Church Benefactors - S8-E20
Visible crew/equipment: During the meeting with the finance committee, a chalk T-mark is visible near Howard's feet when he and Emmett demonstrate their building and safety committee proposal.
The Church Benefactors - S8-E20
Visible crew/equipment: At the start, Andy's trying to get Opie's hair to stay down, and when Aunt Bee appears on the stairs, the moving boom mic casts a shadow at the top left corner of the screen. (00:00:30)
Visible crew/equipment: When Aunt Bee heads out for her first cooking show at the studio, Andy and Opie help load Bee's car, then just as Bee walks around the front of her car, we can see the reflection of various production equipment on the surface of the car. (00:08:00)
Visible crew/equipment: When Aunt Bee comes home enraged at the butcher, a white tape mark is visible on the floor near the kitchen door (it wasn't there in a previous shot).
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)
Visible crew/equipment: At the end, while the foursome sit around drinking pop, in the very last shot of Howard we can see the reflection of filming equipment in the window, before the shot fades. (00:24:30)
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.