Visible crew/equipment: After Barney tells Andy he overheard Dr. Benson and Ellie "talking marriage" while spying on them, Andy heads over to the drugstore. Then, when Barney enters the jail cell, we can see a crew member's fingers (at the lower right edge of the bars, behind the wall) push the cell door closed. (00:19:45)
Continuity mistake: About 20 minutes in, Barney knocks the announcement bell off the door trying to sneak out of the pharmacy, Andy comes in one minute later and the bell rings when he opens the door. You can see the bell swinging above the door.
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.