Deliberate mistake: Luke's dog in the episode is the same dog used in other episodes involving a dog in the plot.
Barney and the Governor - S3-E15
Deliberate mistake: In the closeup of the license plate on the governor's car, the words over and under the plate number are deliberately taped to be unrecognizable. The original words are "drive safely" and "North Carolina."
High Noon in Mayberry - S3-E17
Deliberate mistake: While Luke and Andy are in the house, Barney, Gomer, and Otis are hiding on the porch, and Barney has the idea to turn off the lights in the house, so he goes to the circuit box located beside the windows on the front porch and switches the lights off. The problem is that this circuit box is located on the porch only for this episode as a plot point.
Deliberate mistake: After Barney buys the car from Mrs. Lesh, he and Andy are standing at the steps as he tells her not to look back at the car, and the next shot of Andy and Barney at the front door is the same shot from a few moments prior, but it's been flipped so they would be looking in the opposite direction. (00:09:30)
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.