Telling us where you live can help some. We don’t need your street address, telephone number, and SS#, just enough information to give us a general idea about your climate and which side of the equator you are on. If you stick around on this forum, and I hope you do, you might modify your profile to give us some help with that.
Both metal and shingles will work for a roof. With a coop that size, slope the roof so water runs off instead of standing on top of the roof and slope it in a way that the water runs away from the coop and run, not back into it. You may need a gutter and downspout to get the water away. That depends on where it is situated and the slope of your ground. And have some overhang so you can put ventilation at the top of the walls and still keep rain out.
If you live where you get really hot in the summer, make the coop fairly tall so you have enough vertical room to put in the nests, then put the roosts higher than the nests, then have some clear room above the roosts. Chickens usually like to sleep on the highest thing they can get too. You don’t want that to be your nests since they poop a lot at night. You don’t want poopy eggs. And in the winter if you live where it gets cold, you don’t want a breeze blowing directly on them at night. If you have those ventilation openings at the top of the walls, you want the roost low enough so any wind blows above their backs when roosting.
We keep chickens in so many different conditions and with such different goals it’s hard to generalize on how much space you need. Since you are using feet instead of meters I’ll assume you are in the US. Since much of our building materials come in 4’ and 8’ dimensions, your proposed sizes make sense, a 4x4 coop and 3x8 run. As long as you only have those four hens that will probably work. They should be the same age and there are no roosters involved. But if you lose a hen and replace her, integrating a new chicken in that space could be a problem. I’m a believer in providing as much space as you reasonably can. I find the less I crowd them the fewer behavioral problems I have, I don’t have to work as hard (poop management for example), and the more flexibility I have in dealing with problems. I’d like to see more space but that three foot width restriction makes it a bit of a problem. Maybe just elevate that coop and box in the area under it so that adds to your run area? That gives them some shade to get out of the hot sun too.
It’s not pure square footage that is important. One way chickens have learned to live together is if there is conflict, the weaker runs away from the stronger or avoids them to start with. That’s where the real danger comes in with space. You are more likely to see that type of behavioral problem if they are crowded. The working harder and flexibility is just for your convenience. With all hens the same age you are not likely to see a lot of conflict once they settle the pecking order. That’s why your space will probably work.
Not knowing what your climate is like in the winter, are you planning on feeding and watering in the coop or in the run? We do both so both will work. The feed needs to stay dry so the feeder needs to be out of the rain. You don’t want wet feed molding. If the food and water is in the run, they need access in the winter if the ground is covered in snow. One problem with a smaller coop like you are proposing is that you need to set up the nests, waterers, and feeders so that they don’t poop in them from the roosts. Lots of people do that in a 4x4 coop but it takes some preplanning. One method is to use a droppings board. You put a flat surface under the roosts to catch the nighttime poop and scrape that off as required. That gives you good poop for a compost pile and reduces how often you need to change bedding in the coop. You put the feeder and/or waterer and/or nest under that. But that can eat into your vertical space so preplan. Your nest does not have to be in the coop either. It can either be hung off the side of the coop or a totally separate structure. There are just so many different ways you can do these things it’s hard to be specific. And be flexible. No matter how much you plan, some things will not work out like you expect. You are dealing with living animals, not furniture, and things don’t always work out as you expect.
Welcome to the adventure. I think you will enjoy it.