I see no reason why those screws would not come out - take a utility knife and score around the perimeter of the hinge (so the paint doesn't peel back when removed), then use the utility knife tip or a screw/nail/something to scrape/clean out the paint in the screw head recess so your driver tip holds well.
If it was mine I would likely replace the door with a simple wood frame covered in the HC, so it's always open/ventilated, but the screen door can open to access inside. To close it off such as in winter, I'd make a fully removable solid baffle so I didn't have to deal with more hinges on such a small opening. I'm thinking I'd use rare earth magnets to hold a panel on there. I've made several things that attach with magnets and I really like the round ones with the screw hole in them.
As for your nest boxes - I think the simple straw nest in the corner would work just fine. My hens have access to 6 nest boxes, yet some days none of the hens use them and instead make a nest in the bedding to lay in the corner of the coop.
If it was mine I would likely replace the door with a simple wood frame covered in the HC, so it's always open/ventilated, but the screen door can open to access inside. To close it off such as in winter, I'd make a fully removable solid baffle so I didn't have to deal with more hinges on such a small opening. I'm thinking I'd use rare earth magnets to hold a panel on there. I've made several things that attach with magnets and I really like the round ones with the screw hole in them.
As for your nest boxes - I think the simple straw nest in the corner would work just fine. My hens have access to 6 nest boxes, yet some days none of the hens use them and instead make a nest in the bedding to lay in the corner of the coop.