Ok Ill try to remember and answer you questions, But you have to do me a favor, I would love to see Pic's of the rest of the barn the outside also, Don't matter the shape I love old buildings.
1. The screws and battons on the hardware cloth is a good idea, I used Heavy Gauge Fence/poulty staples that you put in with a hammer, Do you use the floor upstairs? if so you may want to cover it with plywood, thats not a coop issue just figured be easier to store stuff or what ever.
2. The Door gaps, what I would do is frame out for a Screen door like we talked about wood Frame, Hardware cloth over it Ill post a pic of what I am talking about very easy, it'll make the door way a little smaller but you don't need to use big wood just needs to be sturdy and I just use a regular barrel bolt lock but you can use what ever as long as a raccoon can't pull on door and open it.
3. Nesting box is to high, I would lower it closer to the ground say maybe 10 Inches or so to the bottom, and that shelf will get dirty, you can leave it but maybe put something on top to make it easier to clean.
4. The roost are fine but I would add two more boards, one in front and one in between the two you have now, so it is easier for the birds to get up and go from one to the other, some heavier breeds don't fly that well. Since you still have time I would build some kind of chicken droppings collector under the roost, people use sand or Sweet PDZ and they use like sifter it makes it easier to clean out and using a shifter you won't lose your sand or what ever, you will be amazed at how much they go at night and trust me a couple hours of work now and not much money will save you a lot of time and money cleaning up and replacing the bedding under the roost.
5. The Pop door, It may have been aspestos but I think it was probably just regular Blown in paper Insulation, I myself would have put a mask on and kept the pop door down low, but I understand if you don't.
Looking good.