The Taylors in Hollywood - S6-E8
Revealing mistake: While Andy, Aunt Bee, and Opie are at the Hollywood studio watching a scene being filmed, when Andy mistakenly stands up and interrupts the filming we can see the wall and shelves behind the director, and it should look familiar because it's the wall and shelves from Floyd's shop in Mayberry.
Revealing mistake: When Andy arrives at the studio for the publicity shots, while he's speaking to Mr. Michaels note that behind them we can see the Taylors' Mayberry kitchen cabinet/counter and a wall with their distinctive gold/silver design wallpaper. (00:03:50)
The Taylors in Hollywood - S6-E8
Revealing mistake: When Andy, Aunt Bee, and Opie are visiting the studio set while the scene between fake Aunt Bee and fake Andy is being filmed in a fake Taylor living room, note that behind the real Taylors we can see bits of the interior of their Mayberry house scattered in the background offstage. Some things to look for include their living room's stone wall with the mantle, their two reddish chairs (one with the fringe at the bottom), and a wall with their distinctive gold/silver design wallpaper. It's particularly funny when Opie looks at the "fake" living room set and comments, "Is that our house? It sure looks different!" while we the viewers can see the "real" Taylor set behind him. (00:20:35)
Revealing mistake: When the bus pulls up for the Taylor family to leave for Hollywood you can see behind the bus a California freeway with cars moving in both directions. I don't think that Mayberry had a freeway right outside town. The scene is fixed when they return in "Aunt Bee on TV." There has been a tree put there so the road wouldn't show.
The Legend of Barney Fife - S6-E18
Revealing mistake: When Andy and Goober enter the Taylor house Andy calls for Aunt Bee, and just as she walks out of the kitchen we can see the end of the set wall inside the kitchen. (00:14:20)
The Taylors in Hollywood - S6-E8
Revealing mistake: While the Taylors are watching the scene being filmed of the attempted jailbreak Gavin MacLeod plays the sheriff, but while he's fighting the Calhoun brothers it's the actor's stunt double who's actually doing the fighting. (00:17:45)
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.