Driving in Alsace in December

We were there this year around the same time and it was somewhat rainy but roads were fine, just like any other time of year. We drove to Chamonix on December 11 just after a lot of snow had dumped and I fretted about the drive. I need not have worried. It was perfectly fine the whole way. The town of Chamonix had a lot of snow in the ground but all but the smallest streets were passable.
Unless there is some freakish event, you don't have anything to worry about. Enjoy! We loved the markets in Colmar.

The French and German laws about winter tyres/tires are different.

One of the wonderful things about the river Rhine running along the east border of Alsace is the easy way there is of crossing the river and being in Germany with all it has to offer.

Likely, however, if you car was rented in France it won't have the mandatory winter rubber required in Germany.

Re the markets, some of the smaller village markets may only be open on weekends, so plan accordingly.

I wouldn't use Strasbourg as a base with a car, it's a very busy place for its size. I'd choose one of the wine villages further south, then park and ride into Strasbourg. On the outskirts, we once parked at Robertsau Boecklin P&R, then walked through a park close by before walking into town. You can also park near Selestat train station.

Back in December 2019 we based in Colmar, France and had a car to visit Christmas markets along the Route du Vin as well as in Germany and Switzerland - no driving or weather problems.