Other mistake: During the scenes at Elmo's shop, where Aunt Bee's pictures are developed, it's obvious that this is actually Emmett's Fix-It Shop. We can see the window of the shop next door through Elmo's doorway, and recognize it as the store next door to Emmett's shop which has "Furniture, New and Used" on its front window.
Other mistake: Opie accidentally drops and breaks the bottle of perfume and doesn't realise it's colored water, a bottle of perfume would smell for days.
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.