Visible crew/equipment: While Andy's in Opie's bedroom when Opie's thinking about running away, the moving shadow of the overhead boom mic is visible, first on the wall behind Opie's headboard and then on the door. Also note, when Andy shows Aunt Bee how to hold the baseball bat and moves her into position, we can see a long white tape mark on the grass (since it's possible that Andy placed the tape, however unlikely, it's being noted here).
Visible crew/equipment: While Andy and Barney are standing beside Bobby Fleet's vehicle, Andy mentions that it's too bad they're all not staying overnight so they could hear Jim play, and when it cuts to the next shot we can see the reflection of the overhead boom pole and mic moving on the surface of the vehicle.
Visible crew/equipment: After Opie tells Ellie everything that Andy told Aunt Bee, Ellie tells Opie that she's on the trail of a baboon, and when it cuts to the next shot of Ellie coming down the stairs we can see the chalk T-mark on the floor where she stands.
Visible crew/equipment: After Ellie tells Barney that she'll be going to the dance with him, when it cuts to Andy sitting on the porch we can see Aunt Bee's tape mark before she walks over.
Runaway Kid - S1-E6
Visible crew/equipment: After Andy talks to George's father on the phone, he gets up and stands near the fireplace while explaining to Opie why he broke his promise and called George's father, and when Opie's on the couch asking if he can run off sometime, we can see the white tape mark on the rug where Andy was standing moments ago.
Visible crew/equipment: While Andy and Otis are talking about women Barney opens the door and rushes into the courthouse, and behind him we can see a crew member's hand (looks like a woman wearing bracelets) catching the door as Barney swings it shut.
Visible crew/equipment: After Barney tells Andy that the stranger may be a foreign spy Ed Sawyer walks into the courthouse, and behind him a crew member's hand can be seen catching the door just as the door closes.
Visible crew/equipment: When Andy walks into the barber shop while Barney's complaining to Floyd about his sideburns, we can see the white T-mark on the floor at the bottom of the screen, as well as chalk marks where Andy's standing.
Visible crew/equipment: At the end, when Andy opens the door to Floyd's so Ed Sawyer can get a haircut, the set's large spotlight is reflected on the door's window.
Visible crew/equipment: While Floyd's napping in the barber chair, Andy's reading the newspaper, and when Barney walks in the actor's two curved chalk marks are visible on the floor, where he will stand.
Visible crew/equipment: While Barney's in the barber chair, Floyd starts talking about how he'd tried living at the back of the barber shop, and when the shot faces Barney we can see the reflection of a crew member's legs moving around - the only ones in the shop are Barney, Floyd, and the two men seated by the corner.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: When Andy sees that several stores have made 'Hollywood' changes and then walks past the two men wearing suits, it cuts to Andy walking up to 'Hollywood funerals' and we can see the reflection of a crew member pushing the camera dolly with other crew members.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: Before the producer's arrival in his car, when Barney's telling Andy that it'll serve him right if they don't put him in the picture because Andy's not wearing his new uniform, we can see the moving shadow of the boom's hinge and mic on Andy's shirt.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: Near the end, after the producer tells the townspeople to get back to normal and out of their trick or treat outfits, when the crowd disperses we can see some curved chalk lines on the ground near the producer's feet, to mark actors' positions.
Visible crew/equipment: At the end when Aunt Bea enters the courthouse, she doesn't close the door completely. You can see a hand from outside grab the door, and close it very slowly. (00:23:20)
Visible crew/equipment: When Wilbur's in his car he thanks Andy and the rest of the men for all the shoe sales. As he drives off, the reflection of filming equipment is visible on the surface of the car, and when the car is offscreen multiple chalk marks are visible on the sidewalk, at the feet of the actors.
Visible crew/equipment: When the men have the misconception that Wilbur is a TV show talent scout, while they're at the barber shop talking about it, just as Floyd moves his legs to get up from the barber chair we can see the white T-mark at Jason's feet.
Visible crew/equipment: After Aunt Bee plants the seeds of suspicion in Andy's mind about the shoe salesman, when Andy and Barney cross the street the reflection of the moving boom mic is visible on their squad's windshield, just as Andy walks off to the hotel.
Andy and the Gentleman Crook - S1-E21
Visible crew/equipment: When Aunt Bee is in the kitchen preparing to take a tray of food to the prisoner she momentarily walks out of the kitchen through the swinging door. As she pushes the door open, a crew member's hand and arm can be seen behind the door.
Visible crew/equipment: After Barney tells Andy he overheard Dr. Benson and Ellie "talking marriage" while spying on them, Andy heads over to the drugstore. Then, when Barney enters the jail cell, we can see a crew member's fingers (at the lower right edge of the bars, behind the wall) push the cell door closed. (00:19:45)
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.