Visible crew/equipment: When 'FBI Agent' Jenkins congratulates Andy and Barney on behalf of J. Edgar Hoover they're all standing near the door, but when it cuts to Jenkins turning around and they all walk over to the desk, we can see the two curved chalk lines on the floor where Jenkins' feet were positioned, as well as the curved chalk marks on the upper step where the mayor was standing.
Visible crew/equipment: While Andy and Barney are standing beside Bobby Fleet's vehicle, Andy mentions that it's too bad they're all not staying overnight so they could hear Jim play, and when it cuts to the next shot we can see the reflection of the overhead boom pole and mic moving on the surface of the vehicle.
Visible crew/equipment: When Opie's up on the ladder feeding the birds, in the overhead shot of Andy and Barney the mic's cable running up under Barney's pant leg is visible.
Barney Mends a Broken Heart - S3-E6
Visible crew/equipment: While Andy and Barney are with Daphne and Skippy at the Tip-Top Cafe in Mount Pilot, when Daphne's boyfriend walks in we can see the actor's chalk T-mark on the floor.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: When Andy sees that several stores have made 'Hollywood' changes and then walks past the two men wearing suits, it cuts to Andy walking up to 'Hollywood funerals' and we can see the reflection of a crew member pushing the camera dolly with other crew members.
Visible crew/equipment: Just as the first shot opens of Andy walking down the block we can see the moving reflection of a crew member wearing a short sleeve, button down shirt, who's presumably sitting on the camera dolly as it rolls along keeping pace with Andy.
Barney and the Governor - S3-E15
Visible crew/equipment: After Barney has given the governor's chauffeur a parking ticket and the chauffeur drives off, just as Barney starts to say that he's not wearing the uniform to win friends, we can see the reflection of the boom mic at the top left corner of the screen.
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)
Visible crew/equipment: After the boys meet Flip Conroy and talk about the football team, when they all disperse and Opie starts walking away the T-mark is visible on the ground.
Visible crew/equipment: After Andy and Barney walk out of Floyd's shop, when Andy makes a crack about Barney's ears being uneven Andy walks offscreen, then Barney shouts at him as Andy's presumably already down the block, but in their reflection in the window behind them we can see that as soon as Andy walks offscreen in this closeup, he actually stops and leans over where we see the camera's located.
The Darlings Are Coming - S3-E25
Visible crew/equipment: During the scene where the Darling's truck has overheated, Andy walks up from the back of the truck. As he comes into view you can see what appears to be a microphone cord hanging out from under his pants leg.
Andy and the Gentleman Crook - S1-E21
Visible crew/equipment: When Aunt Bee is in the kitchen preparing to take a tray of food to the prisoner she momentarily walks out of the kitchen through the swinging door. As she pushes the door open, a crew member's hand and arm can be seen behind the door.
Visible crew/equipment: After Ellie tells Barney that she'll be going to the dance with him, when it cuts to Andy sitting on the porch we can see Aunt Bee's tape mark before she walks over.
Visible crew/equipment: When the partners of the two men locked up in the jail drive up to the courthouse and park, the reflections of two crew members are visible on the car's vent window.
Opie's Group - S8-E9
Visible crew/equipment: After Aunt Bee asks Clara to speak to Opie, in the interior shot just as Clara walks into the house we can see the two tape marks on the floor, where Clara and Opie will stand.
Opie's Group - S8-E9
Visible crew/equipment: After Opie leaves Emmett's shop it cuts to the Taylor dining room as Opie is walking away from the server with salt/pepper shakers, and when he sits down we can see the actor's tape mark on the floor by the server. (00:06:25)
Visible crew/equipment: At the start, while Aunt Bee is ironing the bowling shirt, a chalk T-mark is visible on the floor under Opie's chair.
Visible crew/equipment: While Aunt Bee, Andy, Barney, and Gomer are on the porch talking about the band, just as Andy says, "We're talking about gettin' up a band concert by tonight," the chalk T-mark is visible on the floor by Andy's feet. (00:12:25)
Visible crew/equipment: While Andy and Barney are hiding behind the car, when Mrs. Lesh hits Jake in the stomach with her umbrella, we can see two curved chalk actor's marks and two chalk T-marks on the floor. (00:22:10)
Visible crew/equipment: At the start, just as Mr. Hampton walks into the office and pushes the door closed, we can see a crew member's hand reaching through the doorway to help close the door. (00:00:55)
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.