My run does not have a solid roof, it does have a small (8x8') portion under the coop. We get a lot of snow so I shovel a good bit(~20 x 8') as well as a path down the entire 50' length(mostly to be able to clear the mesh roof if snow sticks).
I've never free ranged so their confinement area is the pretty much the same all year around. My coop is big enough for my flock not to be crowded during terrible winter storms when they can't go out into the run. I planned it that way and over the years have learned how many birds are comfortable in there.
I've never used food/treats as 'boredom busters'.
Toys? Pfft.
Chickens don't get bored, their keepers do.
If/when it gets icy I'll toss down some shavings or sand.