Rural family here also, so not only do we have a lot of winter gear, but a lot of muck and mud as well. We have a dedicated mudroom, with slate floor-easy to absorb the snow/water and clean mud. I use very large baskets to store the extras, ie snowpants (multiple, black pairs for everyone, rain pants, etc. I also have duplicates of gloves, mittens, scarves, neckwarmers, etc. kept in the baskets. Each family member gets one basket. They are stored on a wall shelf that we installed (old barnboard). Below that we have a very long row of industrial (ie farm) hooks-each family member has three or four, on a long barnboard, screwed into the wall. hard to describe, but it's kind of pottery barn-ish. We hang snow coats, snow pants, insulated rain gear, etc. on these hooks, and they are far enough off of the ground to accommodate the length of the snowpants.
On a wall I have hanging what is essentially a heavy duty shoe hanger, found at a craft fair, made of upholstery material in a pretty fabric. Each family member has a row of pockets, and the pockets hold mittens, gloves, hats and the instant hand/foot warmers that you can get for really cold days. I don't set out all of the mittens, etc. for the season at once because they'll get lost. The pocket organizer helps us keep to one or two pair at a time.
We also have a bench with a flip top lid in which we can store scarves and other misc. I have nice, but industrial, very large sized indoor/outdoor rugs on which we put our muddy or wet boots. I would love to add a boot dryer to the mix as well. I keep a large laundry basket in the mudroom for wet/muddy clothing to be tossed into and carried to the laundry room.
My mudroom isn't heated either (old farmhouse), but that can't be helped. If it's super cold I bring things to the woodstove to warm up, for instance early school mornings, but otherwise we tough it out.
We are the most disorganized family ever, so I needed this plan to get through the winters!