How about setting all corners on cement pavers( no power tools )? Might help with cleaning . My coop is raised but not big so I just crawl in on a dedicated rug runner when I need to clean every 10 days or so.
1 How handy are you ?
2. I don't think 2x2 are heavy enough for weight of snow. I would go bigger and safer with 2x4s ( not that much more money. )
I agree coop is very small so it might wise to add on if you really want 4- 6 hens or maybe go for a smaller breed ? But you've picked some...