
Visible crew/equipment: While Andy and Otis are talking about women Barney opens the door and rushes into the courthouse, and behind him we can see a crew member's hand (looks like a woman wearing bracelets) catching the door as Barney swings it shut.

Continuity mistake: After Ellie tells the Taylors that she's withdrawing as candidate, when she and Andy head over to the rally at the hotel they pass the church, and in the long shot a large sign has suddenly appeared at the front of the church.
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.