Visible crew/equipment: After the squad car is hit from behind Andy rushes over to Goober, and a T-mark (it's a bit darker than the color of the dirt road) becomes visible on the ground (just as the woman gets out of her car) at the bottom of the screen, right behind Andy's feet. (00:01:25)
Visible crew/equipment: Just as Aunt Bee and Opie pull up in front of Wally's station, the reflections of crew and equipment are visible on the surface of the car. (00:06:00)
Continuity mistake: When Aunt Bee and Opie drive up to Wally's station to get gas, the car's vent windows are either open or closed depending on the angle of the shot. (00:06:05)
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.