I think that shed is gonna make an awesome coop! (Don't forget to add some extra ventilation- you can never have too much) Your little coop is almost exactly the same as my old one. I kept them inside that tiny run area for like 2 weeks before I couldn't bear them being so cramped... then they had all day access to my backyard... which got destroyed in about a year.
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So then I had to move it to the side of my house where they could have their own space all day and not eat all my plants! Kind of like your run area you're building. Except my "run area" wasn't covered and they hopped up on the roof and then hopped over the fence to the neighbors yard.
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Chickens are a constant learning curve, I think mostly you have to figure it out as you go.
That little coop didn't hold up very well, so I just recently built a new one that still fits in the same area. And now they can't reach the roof!!
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So anyway.... It sounds like the winter setup you're talking about will be fine. And if you plan on moving the run eventually, I wouldn't bother with burying the hardware cloth. After 2 years that stuff will start breaking down/rusting and will lose it's strength as a barrier. Another just as effective protection is making a hardware cloth
apron about 12-24" wide that surrounds the perimeter. I used metal garden pins to fasten it to the ground then just covered it with gravel or wood chips (depending on where it was located). Digging predators won't think to start digging so far away from what they're trying to get under. And if it does start to rust, it'll be much easier to replace.
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Oh, and one more idea I learned: I don't know if you can see in my pictures, but I built my structures on top of treated 2x6s made for ground contact. That not only helps preserve my actual structure, but also makes the run space deeper so it can hold in a lot more bedding (ie. deep litter, shavings, leaves, grass clippings etc). The composting action brings in bugs and worms, lots more fun for the chickens to scratch through.
(Sorry I'm so chatty)