Revealing mistake: When Andy tells Opie not to drop by the courthouse anymore after school, in Andy's closeups the shots are flipped - note the part in Andy's hair, the top of his badge, and his backwards buttoned shirt.
Revealing mistake: During The Manhunt in the woods, Andy, Barney, the trees, etc., cast long shadows due to the overhead sun, but the problem is that we also see Andy's and Barney's shadows being cast in another direction as well, due to the set lighting on location.
Revealing mistake: When Barney and Andy are searching the woods for an escaped con, they stop to rest and you can see the light level of the woods go up and down as the crew tries to adjust the reflector screen for the desired lighting.
Mayberry Goes Hollywood - S1-E13
Revealing mistake: When Andy is listening to the Hollywood producer talking about how he wants the town back the way it was before, over Andy's shoulder the "window" in one of the stores is blowing in the wind.
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.