The first year I would put the spare shoes in a pack I carried on a homemade rear rack.
I have to agree about the cold hands but it never took that long to start with. The first year I was always been a bit of a whimp and I was only going out to run errands on the nicer days. If it was snowing even threating of snowing I would stay home. Now I've grown up and don't mind riding in the snow like I use to.
Most times the tap and click concept works sometimes I found just riding and letting the snow 'melt' or whatever worked as well. I have carried smaller items with me in the past to clean out the cleats when nothing else works.
I have a steeper driveway, going up to the road. It's not paved so I don't even ride up or down it during the summer months. Since I live on a side road, not a highway sometimes it doesn't get cleaned up as fast as the highway does. Sometimes when it's a light dusting like a couple of days ago they don't even bother to come out and instead just let Mother Nature take care of the roads for them. It's not a heavily travelled road so you don't get the cars going over the snow to help melt the snow like you do on the highways. Walking out to the end of the driveway can still leave me with snow on what used to be pavement until they came out and ground it down to a stone/dirt road early on this past summer. If you get wetter snow versus the light fluffy snow then it will stick to the cleat. Fortunately my driveway is only 100 feet, roughly, from the highway intersection so I can walk up to where the highway is more quickly/better maintained and be able to tap on either the pavement or the pedal and get the snow out, most of the time.