Other mistake: While Andy and Helen are rearranging everything in the squad car's trunk to make it all fit, Helen picks up the large thermos and written on its bottom we see "Reggie personal." Reggie Smith was property master on set. (00:12:45)
Visible crew/equipment: After Goober has deliberately removed the distributor cap from Andy's squad car, when Andy slams the hood shut we can see the reflection of the filming crew on Floyd's window. Then when Andy and Helen drive away in Goober's truck, the microphone pops up into view at the bottom of the screen. (00:17:20)
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.