As a teenager I rode in everything. I lived at 9000 feet in Colorado.
I usually rode bareback - not only do you get to share warmth with the horse, but it also leaves a smaller sweat spot to dry out later. You can use a cooler for that. Nice not having to worry about drying out the girth area, either.
I rode my mare in snow up above her belly at times, for short distances. She would lunge through it (the horse does need to be well conditioned for that).
Barefoot is preferable for riding in the snow, to prevent the balling, but I rode so much those days that my mare's hooves would wear too quickly, so she HAD to have shoes on. My farrier had some pads that had a downward bulge in the middle. He'd put those under the shoes and they would prevent the balling.
These days I don't ride in the winter that much - mostly because I'm more of a wimp (adding 15-20 years does that to you...), but also because where I live now has more ice. We rarely had ice where I grew up, it was just fluffy snow, somewhat packed snow, and then melted. Ice is evil.