For the height you will need to figure the angle you will need for snow to slide off . You could put osb ply wood down then some tin roofing so the snow would slide off the tin, but still have open access to the run and have the pen graded to drain away from the stall .
The rafters on my run-in were strapped then metal roofed. No OSB underneath altho it is noisy during a hard rain. The horses won't stay in then anyway, they move out into a open area to they can see all around.
4x4s would work in that size of shed. They would just need to be spaced closer together than if you were using the 6 inch size. I assume you're going to do a pole shed so you just need to make sure the 4x4s are treated so that they don't rot when they are in the ground.
P.S. I said earlier my shed was 12 ft high at the open end - I measured it today and its 11 ft. Just an fyi as you were thinking you wanted to go with a 10 ft opening.
Ok great! My dad didnt want to set the posts into the ground. He wants to use these instead;
Can you get treated railroad ties. The real ones, not the garden variety. What they call switch ties, the ones for the crossings are longer. I had a load of sand set down, set the ties on that then added the framing of 24" centers. I used rough cut so they were bigger than kiln dried 2x4's. The walls are 8' high plus some extra height from the ties. The frames were lag bolted to the ties. I used proper rafters or trusses (I'm not up on the terminology) because of the snow load they have to withstand.