Visible crew/equipment: While waiting for Barney to show up at the surprise party, Opie looks out the window and spots Barney, then when Aunt Bee and Thelma Lou take their places we can see both actors' marks on the floor - the chalk T-mark and the curved chalk mark.
Continuity mistake: After Eddie Brooke offers Barney five dollars and asks to be let go, it cuts to the State Police car passing the "drugs prescriptions" storefront, then Barney and Eddie trip and scuffle for two consecutive shots, and when it cuts back to the State Police car they haven't even passed that storefront yet. Additionally, that storefront is only two stores down from the coffee shop where Barney and Eddie are.
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.
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.