Revealing mistake: At the end, in the shots of the kids dancing in class, the width of the wall between the shelf/coat hooks and the perpendicular wall with the door has bizarrely increased by 1-2 feet, but that extra width is gone in previous and following shots. Presumably they needed extra space for all the kids to dance so they moved that left wall.
Revealing mistake: When Barney blows the dog whistle to call Blue from the back room, the dog comes running and knocks Barney on his back, and we can see a bit of the cushion that's under Barney's back - it's visible within the shadow under him.
Three Wishes for Opie - S5-E14
Revealing mistake: Barney, convinced he can foretell the future, is in the darkened jailhouse, seated between Goober and Floyd at a table. Goober strikes a match and lights a lamp which is directly in front of Barney. The immediate illumination obviously comes from a studio light because the men's shadows on the wall are to the far left of the men, instead of directly behind them.
Revealing mistake: While Barney's explaining to Andy his new interest in realty they both walk to the back room, and just as they pass through the doorway the camera tracks too far to the left, so we can see the edge of the set wall and the rest of the back wall is missing. (The following shot is from the direction of that missing wall).
Aunt Bee's Invisible Beau - S5-E27
Revealing mistake: When Clara and Aunt Bee are talking in the kitchen, in the closeups facing Aunt Bee we see the counter, the cabinet, and the wall, but what we don't see are the curtains, the window, or the windowsill, which all should be on that wall. The fake window wasn't placed on the set wall for Bee's closeup.
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.