Washington State, almost in Canada and not far from Idaho as the eagle flies. Colder and drier than I often think of your state. Thank you for including the city in your profile.
An 8' x 10' walk-in sounds great to me. it should give you good flexibility. 10' long lumber isn't that more expensive than 8' long and you won't need much of that anyway for the walls.
It kinds of depends on how you build the roof. I like a single sloped roof for ease of construction but you will need at least 10' long rafters to get any overhang. That won't give you a lot, especially on the higher side as far as keeping rain or snow out. For an 8' span you'd need at least a 2x6 for snow load. A gabled roof is more complicated to build but you can build it steep enough so snow slides off and you can get decent protected overhangs. I like overhangs as that is a good place to get ventilation. A 6' x 12' coop would not be that horrible to build. The 2' cut-offs could be used to build nests and such.
As far as plans you can look under the "Articles" tab at the top of this page and go to the coops section. Many of those come with plans plus if they are still active you can maybe contact them and ask if they would do something different.
Getting shed plans is a great idea also. Lowe's or Home Depot often sell books with shed plans. If you chat with your librarian they may be able to come up with shed plans. If not on the shelf they usually can get a book from another library. To make a coop out of a shed you add roosts, nests, and a pop door. And usually ventilation.
How predator-proof do you want to make your run? Will it have a solid roof, a wire roof, or totally open? If it has a roof don't forget snow, ice or even leaf load so build it strong enough or really slope it. How pretty does it need to be? For legitimate reasons that can be important, for others not so much. I can see anything from an elaborate build using lots of lumber to something as simple as posts in the ground with just wire sides and an apron to stop digging predators. Any of these may or may not have a roof.