Visible crew/equipment: After Andy hatches the plan to have Aunt Bee and her friends pick up their photos at the same time, in the following shot just as Andy walks into the shop, the shadow of the boom mic is moving beside the doorway, at the top left side of the screen.
Visible crew/equipment: After Aunt Bee's flight lands, when it cuts to the hotel lobby there's rug on the floor extending from the front desk to the stairs, but in the next shot as Aunt Bee and her friends walk into the lobby that rug is gone (this doesn't refer to the small rug behind them). Also note the tape mark on the floor where the bellhop stops the luggage trolley.
Visible crew/equipment: When everyone is at the airport before the flight some chalk T-marks are visible on the floor, such as the one between Andy and Helen when Andy takes the picture of Aunt Bee and her friends. (00:08:45)
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.