Freeze water bottles and put them in the run during the day to help them cool down, keep drinking water cold, and provide a whole lot of shade. It is preferred to lock up your birds at night to avoid predators from getting your birds, and if there are windows or any opening other than the door (which their should be) keep them open to allow fresh air to go in.
And depending on the breed, pullets will begin laying when they are 4-6 months old. Use anything small and egg-looking to put in the nesting boxes so they can follow the example.
And, the heat usually is not a problem. Even though a nesting box isn't as drafty as outside, most hens will prefer a dark, small place to lay their eggs.
Also, I get all of my eggs before ten in the morning, every day. They do not stagger down an hour every day, but that's just me. My birds currently laying are hybrids, so they lay fore often and consitently.