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).
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.