A wet coop or run is a dangerous coop or run. Wet poop stinks and diseases can breed in wet dirt or bedding with poop mixed in. In wet weather you may not have a lot of options in the run, it just may get wet unless you wrap it. Rain blows in from the side, not just from on top. When it rains for a few days I just live with mine.
I agree about doors on your coop or gates to your run. They need to be built and well-supported so they don’t bind. It’s considered to be good practice to build things level as that makes it easier to build doors and windows so they don’t bind. It’s easier to set the hinges properly. Paneling fits corners if the corners are square, otherwise you may need to do some fancy trimming. It’s just overall much easier to build a building if the foundation is level. It has nothing to do with the chickens or their comfort, they won’t really care. It’s about you and your stress levels. Even if you build the coop somewhere flat and install it, I’d still try to get the foundation level. That makes it easier to modify it in the future if you need to, and many of us need to.
I suggest a berm or swale on the uphill side to divert rainwater run-off away from your coop and run. Even on a gentle slope you can get erosion if the vegetation is disturbed or the ground is softened. Yours will be. Diverting the rainwater run-off may create a ditch that gets eroded by concentrating that water in one spot. It’s very common when you do construction on a slope to create erosion problems, it’s just something you have to watch. You do not have much slope so it's not likely to be a big issue. There are ways to manage that if it happens.
Build any roof so water drains to the lower slope so it runs away from your coop and run, not into it.
You don’t have to worry about leveling the run, the chickens will do that for you. Chickens scratch a lot. Gravity moves things downhill. Over time, and not a lot of time in that tiny run, they will move the dirt from the top of the slope to the bottom. You need to keep that in mind when you are building your foundation. Have something solid along the downhill side to keep that scratched dirt from continuing down the slope. On the upslope side the level of dirt right next to the fence will probably drop a few inches. I don’t know how you plan to build your foundation or the bottom of your run, but in something that tiny leveling the entire bottom of the coop and run foundation may be a good thing.
I don’t know anything about your climate, how many chickens you plan or what sexes and ages, how you plan to manage them, or really anything else. What you plan could work out great for you. But I suggest you follow the link in my signature about space. The article gives you things to consider.
Most popular and inexpensive building materials come in 4’ and 8’ dimensions. If you bear that in mind you can often build something bigger at little extra cost and with less cutting and waste. That 5’ dimension looks a little awkward, maybe you are planning on using the cut-off for nests or other things. Just something to consider.
Also I don’t know what style coop you plan to build. I really like overhang at the high and low side of a sloped roof. The roof needs to be sloped so water runs off, otherwise your roof leaks or standing water rusts or rots your roof. I leave the tops of the walls under that overhang open (covered with hardware cloth for predator protection) so I get really good ventilation. No matter what your climate you will need good ventilation.
I’ll say it again, the chickens really won’t care how level it is. But you might. It might be really important to you.
Good luck and welcome to the adventure.
