There's a police radio on the table next to the gun cabinet in the sheriff's office but it's never used. All communications to law enforcement are done either in person or via the ancient candlestick telephone.
Opie carries his schoolbooks with a leather book strap. These weren't used by schoolchildren in the 1960s. They were popular from about 1850-1900.
It's unbelievable that the TV station would allow Aunt Bee to host the cooking show without even a single rehearsal, her only other TV appearance being on the quiz show in season 6 episode 10 "Aunt Bee on TV." Some of her gaffes could've been avoided.