Revealing mistake: While Goober is up in the tree with the fake owl, when Andy and the Darlings leave the house they are standing on the porch, and after seeing the 'second omen' the Darlings leave the porch and head to their pickup truck, but what's funny is that we can see their reflections in the window behind Andy as they all walk off camera, turn around and just stand there, and we can still see those actors' reflections in the shots facing Andy and Aunt Bee even as the Darlings supposedly drive off.
Visible crew/equipment: When all the Darlings walk out of Andy's kitchen with a bag full of Aunt Bee's muffins, the actor's tape mark is visible on the floor, where one of the Darling boys looks down and stands. (00:20:25)
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.