Germany or Italy during Christmas

We lived in Germany for four years and loved the Christmas markets, but I believe they close about the time you plan to go to Europe.

One thing, I hope you like cold weather. If you go to Germany, take a nice warm overcoat, gloves, hat and warm shoes.

Italy would be warmer, but still cold.

You need to go at least a week earlier

Most markets in Germany will close this year on the 22nd. Check the websites of the cities to be sure.
The 24th is the celebration and the 25th and 26th are official holidays.

I agree, you are going a little late at least for any christmas markets. I lived in Gemany 4 years as well, in the mid 80"s (the celebrations don't change). italy I don't know? The Vatican will have it's famous Christmas Mass. You may still be there for All kings day (Jan, 6th). An important Catholic holiday. Of course there is NewYears as well.
Travel well!

In Berlin you will find Christmas markets which are open between the years.

Most Christmas markets in Bavaria close latest Dec 23rd - one exception in Berchtesgaden (source).

As above, if you travel on the 22nd you will miss all the markets in the towns in Germany you listed. This is a bad year, due to the 25th and 1st being on a Wednesday.

As to cost, having just researched it for family planning to visit, you will find most hotels in those cities already filling up and the average cost of a room at approximately twice the summer rate. The hotels I use in Nuremberg and Munich (which I have used several times this year already) are running $400-$600/night from 1 DEC on, if they still have rooms.