Continuity mistake: At the "Hen Party", Jeff Pruitt says hello to Thelma Lou and she states she is going to get coffee, then goes to the kitchen. However, when Jeff walks across the room and looks back, Thelma Lou is back in the same area. She never walked back from the kitchen.
Continuity mistake: In the initial episode, and Runaway Kid in season 1, Barney is a cousin of Andy's. In subsequent episodes, they are unrelated.
Continuity mistake: Scenes shot at Andy's porch have significant differences depending on the filming location, as there are two sets of Andy's porch. One is just the facade of the house in the 'town of Mayberry' in the studio's 40 Acre backlot; and the second is located in the studio's soundstage. Note in some shots the door's sidelights have vertical muntin bars, the screen door has a white handle, whereas in other shots the sidelights' muntin bars are gone and the screen door handle is black (screen door's gone season 5). Also, note the significant difference in the width of the wall between the parlor windows and the door, the height of the parlor windows from the floor, etc.
Continuity mistake: The fuel pumps at Wally's gas station (regular and super) switch sides between numerous episodes.
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.