Continuity mistake: When Andy locks himself in the jail cell, the key is lying flat on the short wall, but after Barney locks himself in the other cell, the key has turned itself upright on its side.
Other mistake: After Barney gets annoyed over Andy's lack of confidence in his ability to be a sheriff, Barney gets on the phone with Sarah and asks to be connected to the mayor of Greendale, then it cuts to an exterior shot and the parked car has plate number "DC 269," the same as other vehicles in some earlier episodes.
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.