People who are from, live in or have lived in Germany/Austria will need to chime in here. I lived in Germany only for 3 years, 1982-1985, but here's my take on what you asked and said.
He has dreamed about sleigh rides...Yes, you can do that, especially in locations near the Alps. It's a lot of fun and magical being in an open sleigh being pulled over the snow by a horse or horses with the bells ringing. It's not anywhere close to warm, though.
Christmas markets... Yes, they are charming. I still have my beautiful red and gold angel from the Christkindlesmarkt in Nuremberg along with fond memories of brotchen and bratwurst and gluhwein and lebkuchken and coffee.
The beauty of the holidays in Europe...Much more low key. Not as commercial or as decorated up as the US.
Me, the negative/realist, thinks of being freezing cold...Yes, it will be cold. You need to dress for it.
Icy/snowy roads...Yes, I drove on some regularly to and from work and rode on some in buses on group trips. On one skiing trip, the bus taking us to the resort got stuck at the bottom of an icy mountain road. After standing around and stamping our feet, most of us walked up the road to the place. White snow. Bright moon. Nobody fell off the mountain. We all got there about 6 hours later than anticipated and the bus arrived considerably later than we did. No food, nothing available for this large group. And no extension of breakfast hours the next day either.
What to do beyond Christmas markets?...As others have said, there will be concerts, especially in churches. Attending a service is wonderful even if the church isn't heated and you don't understand German. My personal favorite is going to a spa with indoor sauna and wave pool.
We don't like huge crowds, so we would prefer smaller markets...Even at a famous market like the Nuremberg one, you can avoid huge crowds by going at somewhat off times when there aren't so many people, but honestly, part of the fun is all the people and little kids. Also there will be crowded stores that are open later for shopping during the season. I remember getting in lines to ride the escalators at a department store.
Is it reasonable to be out driving this time of year?...If you are not used to driving in snow and ice, I wouldn't do it. Check out public transportation. I'm sure you could get most anywhere that way. That's a big advantage we often forget about in our car-obsessed world here.
Can you get snowed in?...Yes, we got snowed in at the top of Grossglockner in Austria in late June! Even the super, duper 4WD folks didn't leave until the parking lot and road up had been plowed and was mostly melted.
What do you do there?...There will be the activities already mentioned, but don't expect anything to be happening on Christmas day. In Stuttgart, absolutely nothing was open. I was visiting a friend and we made tacos, then went on post to the bowling alley and had hot dogs.
Just walk the villages and cities?...That sounds about right, but depending on where you stay, and who you might meet there, that could be a lot of fun. Keep in mind that the days will be short, very short.
Just wondering what kind of cold- deep chill damp cold or dryer cold?...Deep chill damp.
I wouldn't let any of these conditions keep me from going. It will be a beautiful experience, just not a SoCal one.