I always thought about getting some too but didn't for the same reason. Our neighbors are pretty bad with being nosey, I'm surprised I've gotten no complaints about my birds yet.
Chickens can handle the heat, especially those breeds. They don't crow because they are in distress, it's a territorial call. If they are crowing more it's not because they are over heating or being cared for improperly. It just shows they are healthy and strong.
Here are some plausible...
I'm more concerned that you are watching and photographing someone else's children in their own backyard than I am concerned about the chickens not having a door on their incomplete coop.
They must have food, water, and shade. They would be dead otherwise. I'm sure the chair provides a square of shade big enough for one rooster through the day. Even if it is moving around, their attachment looks long enough to reach anywhere the shade spot would go. The coop they are putting up...