When Andy and Barney first see Gomer on the courthouse roof, they're standing on the sidewalk. Then they're standing in the street, talking about Gomer's feet before Gomer drops his gun from the roof. When the gun lands, Andy and Barney are back on the sidewalk, then they're in the street again as Barney yells at Gomer, then back on the sidewalk when Andy picks up the gun.
When Gomer first drops his gun from the roof, it points left but lands on the sidewalk pointing to Andy and Barney's right. Seven seconds later, when Andy picks up the gun, it points left again.
Repeated reference is made to the "State Police." However, no such agency exists in North Carolina, where the series is set. The state law enforcement agency there is the North Carolina State Highway Patrol, which has gone by that name since its 1929 founding.
Just before Barney jumps out from behind the crate to surprise the two cops in the alley because he thinks they're criminals, the director can be heard shouting "Don," to give actor Don Knotts his cue when to jump out.
When Barney jumps out to surprise the two cops that he thinks are crooks, he is in a dead-end alley. There is no way for him to get past and behind the two cops without them seeing him.
At no point do the State Police share photos with Andy and Barney of the career criminals at large they are all aiming to capture, nor do they inform Andy and Barney that they are coming. Andy and Barney learn of their impending arrival from the radio newscast.
When Andy puts Doc and Tiny back behind bars after their second escape attempt, he hands Gomer's broken gun to Barney. Andy would have to have been behind the courthouse to see and retrieve the broken gun, then walk around the building to enter through the front door rather than enter from the back. There's no reason for Andy to do this other than the script's demand for him to fortuitously catch Doc and Tiny at the front door.