After they enter the cave, Barney tells Thelma Lou that bats get tangled in your hair and lay eggs in it. Bats are mammals, like humans and give birth to live young. They do not lay eggs.
In the final scene, which takes place inside the Mayberry Courthouse, Opie brings newspapers into the courthouse and leaves the front door open, revealing a building down the street: the courthouse! In reality, it's the backdrop normally used for the front view of the Mayberry Courthouse, including the columns on either side of the door.
Barney leaves the picnic grounds to muster people and gather rescue tools. However, he makes no attempt to get hard hats nor do any rescuers wear them.
Andy and Helen hitched a ride back to town after getting out of the cave, yet somehow got back to the cave after hearing of the rescue mission on a radio broadcast. Helen apparently has no car and Andy's squad car would have still been at the picnic site.
So as not to embarrass Barney, Andy and Helen decide to re-enter the cave that just collapsed on them, not the best judgment exhibited by a sheriff or a school teacher. The cave is obviously unstable and Barney and company could possibly trigger another cave-in, yet Andy and Helen recklessly go in to save Barney from further humiliation by the townsfolk.