When Barney tests the camera in the filling station, his mouth is open when he opens the door (and triggers the shutter). When the photograph is developed, his mouth is closed.
When Andy first walks up to the filling station door, one pant cuff is tucked into his boot and the other cuff is hanging loose. But when he gets inside, both pant cuffs are loose.
The way Barney set the string attached to the camera and door *would* have triggered the camera. The door opened to the inside of the garage and the string, strung through a hook on the door frame, would indeed have pulled the shutter switch.
When the picture taken from the camera is revealed to be Barney, as he was testing the mechanism to make sure the camera would take a picture, Andy says it does not change anything and still wants to arrest Jimmy. In fact, it proved Barney's trap worked, since Jimmy only had a key for the front door. The fact that there was only one picture, of Barney, exonerates Jimmy. CORRECTION: The problem with that idea is that the camera is a type which does not have an automatic rewind. It takes only one picture at a time. And since it did not have a flash, Barney did not notice it in order to reset the camera. His picture was purely an accident, not a "test of the mechanism to see if it worked." And another thing about all this is, without a flash or high speed film, any picture taken at night, without any light, would not capture an image.
There was no way for Mr. Carter to know that Andy was not at the courthouse and yet he brought Jimmy to him at the filling station. EDIT: The assumption is that he meant to find Andy at the filling station, if he wasn't stopping to fill his tank or something beforehand.
When Gomer talks to Andy at Wally's, Andy has an unlit cigarette in his mouth. When the angle changes, Andy acts like he's taking a puff, but there isn't any smoke and then he drops it to the ground and crushes it out with his foot.
In every instance when the courthouse door is open, it is revealed that the outside, including the squad car and the building across the street, is a canvas painting.
Gomer states that the robber couldn't get into the cash register because he has the key and keeps the register locked. Later, when Andy and Barney catch Jimmy in the filling station running the cables from the cash register to the batteries outside to shock anyone trying to steal the money, Barney suggests maybe Jimmy is going to blow open the cash register. In the next scene, when the robber is electrified, the cash register drawer is open. No explosion and the robber didn't have a key, yet the register drawer is wide open. Furthermore, if the metal cash register is electrified, the drawer wouldn't need to be open for the electricity to shock a person.
There is never a mention of how the thief got into the trunk or how the stolen items would have been removed from the garage.
Roosters don't lay eggs, but when Andy hears Barney saying "cock-a-doodle-doo" to Juanita, he jokes that he wanted to see if there was an egg on Barney's chair.