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.
Ernest T. Bass Joins the Army - S4-E3
Visible crew/equipment: When the two men are carrying the window pane past Barney and Andy, we can see the reflection of a few crew members' legs.
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: Barney shows up with the concert sign while Andy's sawing wood, and when he and Andy talk about it, the shadow of the boom pole/mic is moving around on the ground, at the lower left corner of the screen. More of the boom's shadow is visible again, when Gomer shows up and they talk about Goober. (00:14:30)
Visible crew/equipment: At the end, when Dr. Breen tells the very tired group, "You all look as serene and relaxed as if you'd just finished listening to a pleasant band concert," in the next shot facing the porch, the shadow of the boom mic is moving around between the two windows above Clara and Andy, at the top of the screen.
Visible crew/equipment: At the end, when Barney's talking to all the boys at the courthouse, the camera angle's looking up in Barney's closeups, and the top of the courthouse set is visible right before Barney mentions "intimidation."
Visible crew/equipment: After Barney brings the "Fun Girls" to Andy's house, when Aunt Bee walks into the parlor there are now three black tape marks visible on the carpet - and those marks weren't there when Barney and the women walked in.
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.
Visible crew/equipment: At the courthouse, while Barney and Thelma Lou are trying to stall for time, in the shots facing Andy and Helen when Andy says, "Steaks, chops, macaroni surprise," we can see the shadow of the boom mic moving around at the top, left side 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.