Other mistake: Right before Aunt Bee wins the five dollar prize at Wally's station she talks about how she's never won anything, not even the church raffle, and right after she wins she excitedly waves the five dollar bill and says, "I've never won anything before in my life!" However, in season 6 episode "Aunt Bee on TV" the man from the IRS stated that Aunt Bee had won $4,850 worth of prizes on the game show, and even though she sold most of her prizes she kept two of them. So her statement makes no sense even for Aunt Bee.
Other mistake: Twice during the episode the actor who plays Floyd is replaced by his body double. The first time is when Floyd's driving to and away from Wally's station in the earlier scene, and the second is later, when Aunt Bee leaves Floyd's shop and we see Floyd stand up through the window.
Visible crew/equipment: When Aunt Bee drives her car up to the gas pump at Wally's station and then wins the five dollars, the reflections of crew and filming equipment are visible on the surface of her car. Also, note that the chrome trim around the windshield is covered with black tape, presumably to prevent reflection. Additionally, during all the scenes at Wally's station, the back end of one of the quonset huts from the "Gomer Pyle, USMC" set are visible, either at the left side of the screen or in the background.
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.