A 4x8 coop with an 8x16 run for ten full sized fowl chickens will concentrate poop in a fairly small area. I understand the coop is intended to be raised but the run will still be wet a lot. When chicken poop saturated ground stays wet it stinks and can be unhealthy. Even if it were dry with facilities that small you will be managing a lot of poop, probably in the run as well as in the coop. That's one of the disadvantages of having smaller facilities, you have to work harder to make them work.
We live in Northeast Ohio, where temperatures range from average lows in the 20's in the winter (although sub-zero windchills happen occasionally) to average highs in the mid 80's (with peaks up to mid-90's) in the summer.
Give your chickens a way to get out of the wind and wind chill doesn't mean anything. Your average temperatures is not where you get into trouble, it's the extremes. When I was in Arkansas I seldom saw temperatures much below zero Fahrenheit. In northeast Ohio you should see temperatures below that. Mine were outside enjoying the weather when it was below zero F.
To me winter sun is not very important. As long as you give them a way to get out of the wind they can handle cold temperatures really well. Summer heat is what can kill them. Mid 90's isn't that bad but they will need shade and plenty of water. When it got above 110* F I had a couple of hens die. When it was in the upper 90's I'd wet the ground in the shade to give them a cooler spot. And plenty of water to drink or stand in. They were hot but they did not die.
My coop and run was about 200 feet from the house. I ran electricity and water to it, all according to code. I hired professionals to do it, can't remember what it cost. I installed a frost free hydrant which was great. Water down there was really convenient for my orchard, especially when I first planted it. That made watering the garden and berries much easier too. In good weather you can run hoses but hoses don't do well around lawn mowers.
Although I had electricity down there I did not heat the water in winter. I'd water in black rubber bowls I got from
Tractor Supply. If I set those black bowls in the sun solar heat would keep water thawed down into the teens. The sun doesn't always shine, especially at night, so I'd knock the ice out of those rubber bowls and refill them as needed, usually once or twice a day. With this type of watering they will poop in the water so you need to change the water every day or so anyway. With electricity you could rig up a nipple system and keep that thawed. Lots of different ways to do things.
My coop and run were downhill from the house but on a very slight rise. I put in a berm and swale on the uphill side to divert rainwater run-off away from them. My 8x12 coop was on the ground so I hauled in enough dirt to raise the dirt floor a few inches. It stayed dry. My run was mostly covered but rainwater would still blow in from the side. When the weather set in wet it would get wet, but being on a rise it drained fairly well. I also had an area about 45' x 60' in electric netting which was where they spent most f their waking time. The poop did not build up that much in the 12' x 32' main run.
I don't know what the lay of your land is in that closer spot. If you have a lower spot near it, a French drain can move a lot of water. It might be worth doing. An 8x16 run just isn't that big. 10 chickens are going to keep it totally stripped of green stuff. Whether you do a French drain or not I'd give real serious consideration to building it up with gravel so it is above the surrounding area. I'd look for rounded rock like river rock or pea gravel so they don't cut their feet on the sharp edges of crushed rock. Rock like that will drain really well. Then I'd cover it with a few inches of a course sand. That should keep it dry. With that many chickens in that small a space the poop is going to build up. I think you are going to have to manage that poop some way, maybe removing and replacing some of the sand. Your nose will tell you how often you need to do that. The "French drains" I've used have been a trench filled partially with gravel. Maybe before you put gravel in that run toss in some of the dirt from digging your French drain?
My main suggestion is that you have a lot of room out there. Use it. I know it is expensive to build bigger, but you seem to have the mindset that you are living in an urban area. From those photos you are not. I think you will be much happier in the future if you build bigger, especially the run. You are going to be taking care of those chickens every day. Make that as convenient an comfortable for yourself as you can.
