Continuity mistake: When Opie shows Andy his arm with the chore list written on it, Andy uses his right hand to hold up Opie's arm, but in the closeup it's Andy's left hand holding up Opie's arm. (00:03:50)
Continuity mistake: At the start, when Andy stops at Wally's station Gomer is telling Jason about his Trout bait, then someone pulls up behind Jason's car and they're bumper to bumper in the wideshot, but in the exterior shot facing Jason there's no car behind him, but that car is back again before he pulls away.
Continuity mistake: In earlier episodes, we are told that Gomer can't do mechanic work. He just pumps gas and air: "There's no charge for air; air is free; we don't make no charge for air." But in this episode, Gomer has become an ace master mechanic.
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.