I haven't done any hunting from horseback, but I can take my best shot.
The snow should be dry and fluffy enough that it won't cage up on a dog's paws, or a horse's hooves. It should be fairly easy to move through it. Of course after a couple of days the snow would start to pack down and stick more.
It sounds like they intended to go after some bigger game, and would bag some birds if the opportunity arose. Nobody wants to share their seat on a horse with 90 pounds of dead antelope for a long ride back home, so they probably had a pack horse or pony just for that. They'd be wise to bring along some food for people, dogs, and horses, since it's cold out, and everyone uses more fuel.
A big danger in "perfect" weather like that is dehydration, which can sneak up on you much easier in cold weather than hot. An insulated jug of hot coffee might be good for humans, but the animals would appreciate some clean snow melted down and heated to near body temperature. So a kerosene camp stove and a bucket or pot would be something for the pack horse to carry on the way out.
And I love that kind of weather, too. My fondest memories are of my father and me, .22s slung over our shoulders, enjoying the bright sun, fluffy snow, and absolute quiet of my grandparents' woods after a light snow the night before. It can't be surpassed for peace and tranquility.
And, here's some references that might help:
Winchester 1886 takedown rifle in 45-90. This particular model looks like it's ultra-deluxe with every possible posh feature other than gold inlays:
To cover the birds-and-antelope situation, here's a typical German drilling (triplet) in 16x16x7.8mm:
The business end looks like this:
These are not items available at the usual Texas small town gunsmith/hardware store. They'd either be brought back from a trip back east, or special ordered. The same things apply for British shotguns and double rifles. Anything that was special-ordered probably had at least a one-year wait on it, even in those days.