Continuity mistake: At the start, Opie and his friend rush to the front door of the same house to try and sell their products - note the house next door (and the fact that it's three houses down from Andy). The kids then run across the street and have the door slammed in their faces, but then it cuts to the woman next door as she walks to her front door - note the house behind her is that first house Opie went to, even though all the boys just crossed the street a few moments ago.
Visible crew/equipment: At the end, when Barney's talking to all the boys at the courthouse, the camera angle's looking up in Barney's closeups, and the top of the courthouse set is visible right before Barney mentions "intimidation."
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.