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).
Emmett's Brother-in-Law - S8-E18
Visible crew/equipment: When Ben accompanies Emmett on his first call to Andy, just as they both enter the courthouse, we can see a crew member's hand outside reach out to catch the door as it closes. (00:12:40)
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)
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.
Barney Hosts a Summit Meeting - S8-E21
Visible crew/equipment: When Andy and Barney start to get into their car, the boom mic is reflected in the front window. (00:10:10)
Aunt Bee's Big Moment - S8-E23
Visible crew/equipment: After Aunt Bee's solo flight, just as the flight instructor walks over and gives Bee the papers as proof that she soloed, the large T-mark (rubber/plastic) is visible on the ground between Bee and the instructor.
Aunt Bee's Big Moment - S8-E23
Visible crew/equipment: After Aunt Bee tells Opie that she will solo the next day, when she sits in the chair to practice her flight maneuvers, the actor's tape mark is visible on the floor beside her feet.
The Wedding - S8-E26
Visible crew/equipment: The night of Howard's party, when the doorbell rings and Howard opens the door for Goober, we can see a crew member's head moving in and out of the shot at the right side of the screen.
Visible crew/equipment: After Andy gets the phone call about Sam's victory, when Sam is pushed to give a speech, a few white tape marks are visible on the floor near the fireplace.
Visible crew/equipment: After Emmett talks to to the diner owner about having the bus stop moved to double his diner business, when Emmett walks down the block, the reflection of the moving camera dolly is visible in the store window.
Visible crew/equipment: When Andy, Howard, and Goober walk into Emmett's shop to tell him that Sam's running for head of City Council, the shadow of the boom mic is moving on the wall behind them, at the top of the screen.
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.