Visible crew/equipment: While Ollie's making up excuses to rush home, he and Nora are saying goodbye to Aunt Bee, and the reflection of the moving boom pole and mic are visible on the car's windshield.
Visible crew/equipment: While Andy's sitting on his desk going through the mail and Barney's sitting on the side desk, the actors' chalk marks are visible on the floor behind them, in the shots facing both of them.
Visible crew/equipment: When Gloria and Frank drive off at the end, just as the camera zooms in on Andy and Helen, the shadow of the moving boom mic is visible at the top, left side of the screen.
The Education of Ernest T. Bass - S5-E4
Visible crew/equipment: When Ernest T. runs across the street to help Helen with her groceries she rushes away from him and heads up the block to Andy, and as Andy and Helen walk toward the courthouse the shadow of the boom mic passes across the white pillar in front of the door.
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.