Three Wishes for Opie - S5-E14
Other mistake: When Andy asks Floyd and Judd about who he's supposedly marrying it cuts to Helen walking down the street, and the grocery store has only one display window, when it should have two windows, and there's a brick wall adjacent to the window which shouldn't be there. (Shot at the soundstage not backlot).
Barney Runs for Sheriff - S5-E21
Other mistake: When Floyd tells Barney that he doesn't want to put Barney's campaign poster in the barbershop window because it spoils the decor, the reflection we see in the mirror behind Floyd is the church's front door and steps. This makes no sense at all because the reflection should be whatever is to the right of Floyd's, but the church is the opposite way - to the left, past the courthouse. The view in the mirror is a painted backdrop, but it's still quite wrong.
Other mistake: In this episode (aired Sep. '64) Barney tells Andy that 5 years ago on May 16 (1959) he joined the sheriff's office, and later they all have a surprise party for Barney. However, in season 4 "Citizen's Arrest" (aired Dec. '63) Andy tells Barney that he found, "The custody receipt for the first revolver I issued you... August, 1953... 10 years ago last month since I swore you in." Andy even mentions having a party with a cake that has 10 candles on it.
Three Wishes for Opie - S5-E14
Other mistake: When Barney uses the gypsy lamp and cards in the back room, in the shots with Goober and then with Opie while they're standing in the doorway, the wall opposite them has a barred window, but in earlier episodes that wall has the door that leads outside to the side of the building.
Other mistake: When Andy and Helen are in the squad car alone after going to the movies with Gloria, both car doors are closed, but in the closeups from outside both sides of the car, note that both triangular vent windows are missing and we can see the doors' rubber seal - both doors are open for these closeups.
The Case of the Punch in the Nose - S5-E25
Other mistake: In the shots from outside the courthouse, when the front door is opened there's a wall where none should be, presumably to prevent seeing inside the courthouse facade.
Other mistake: After the preacher walks past Andy and Sam, when the two men comment that the preacher's sermons "go on" and are "dry as dust," we can tell that all of the store's glass window panes have been removed, presumably to avoid the crew/camera's reflections as the shot pans across the window to Howie.
The Arrest of the Fun Girls - S5-E28
Other mistake: When Andy, Helen, and Thelma Lou are parked in front of the courthouse the lights turn on, and in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups. This happens again later, when Andy and Barney are each apologizing individually to Helen and Thelma Lou.
Aunt Bee's Invisible Beau - S5-E27
Other mistake: After Andy talks to the butter and egg man in Mt. Pilot, when Andy gets in the squad car with Barney all the car doors are closed, but in the closeups from outside both sides of the car, note that both triangular vent windows are missing - the two doors are actually open for these closeups.
Opie Flunks Arithmetic - S5-E30
Other mistake: When Aunt Bee walks out of the grocery/meat store the door's located between the grocery's one display window and the TV display window, but the grocery store actually has two display windows with its door between the two. This was shot on the soundstage, not backlot.
The Education of Ernest T. Bass - S5-E4
Other mistake: After Ernest T. breaks Helen's window Andy shows up, and there's black tape adhered to the chrome trim at the side of the windshield and also the chrome trim beside the vent window.
Other mistake: When the escaped convict takes Barney hostage, they walk away from the lake and Barney drops the piece of paper with the convict's picture. Later, when Andy walks across the grass he finds Barney's cap and the paper lying on the grass, but the problem is that Andy needs to pull up at the center of the paper because it's pinned to ground, presumably so it wouldn't move or blow away for the shot.
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.