The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Helen and Thelma Lou walk into the courthouse and find Andy and Barney dancing with Daphne and Skippy, there are lipstick marks on Barney's cheek, but when Andy gives Skippy her sweater and radio then tells them both to go, in the closeups the lipstick marks are gone, though they reappear momentarily.
The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Daphne and Skippy drop in on Andy and Barney at the courthouse and want to "play jail," Skippy carries the radio and her sweater into the jail cell with her, but when Skippy asks for her ukulele we see the radio and sweater are now on the side desk, under the gun case.
The Arrest of the Fun Girls - S5-E28
Other mistake: When Andy, Helen, and Thelma Lou are parked in front of the courthouse the lights turn on, and in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups. This happens again later, when Andy and Barney are each apologizing individually to Helen and Thelma Lou.
The Arrest of the Fun Girls - S5-E28
Character mistake: Barney states when the Fun Girls were in town before, in Season 4, Thelma Lou didn't speak to him for 10 days. That's not true, they made up the same day.
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.