Not sure how they could peck at it enough to ingest it. Never had that issue.
If you're building your own coop, add the linoleum before you frame up the walls. The walls will sit on top of the linoleum, eliminating the possible issue of the edges curling up if it's cut to fit after the walls...
It will be back, and she will take a chicken with you standing in the yard only a few feet away if it gets a chance.
I'm thinking your hen was bouncing off the run walls and that's how she got cut up.
I'm in upstate NY also, and Yogi has been very active in my area. This guy is big, and took two bags of garbage off our front porch. Backyard is fenced in so the dogs do a good job of keeping him and other predators out.
I'm in NY also. I'd open both gable ends up. Or at least one if you could get some of those transom Windows open. Don't worry about the window being at roost hight. The gals will love it open from spring to fall. Maybe a awning over it if prevailing winds could blow the rain in.
Transom Windows...