Most everything closes Christmas day in Europe. Even those that don't close will have minimal staffing.
My brother will be taking his wife over for 3 weeks over the Christmas/New Years holidays this year. They went over 3 years ago (her first time) and she really wants to see the Christmas markets and other festivities. They will spend Christmas near Salzburg based on my recommendation with plans to take part in the Advent in Oberndorf.
I highly recommend Boutique Hotel am Dom in Salzburg!
In Vienna I have used Pension Wild and been very happy. It's very safe, located between the city hall and a major Gov't building, on a backstreet in a good part of the central city.
I'd like to point out that to really see the Christmas markets you need to do them at night. Daytime is for tourists. Locals come out after dark and that's when the party really starts. In Nurnberg you should also plan to walk the path along the Pegnitz through town and look at the displays in the windows. Bonus points if it's snowing.