Family travel tips for Christmas in Germany?

With a 3yo spend 2 nights in Salzburg so you have a full day for the Christmas market and castle. In Munich go to the Deutsches Museum, it has a full day's of interesting things to see and do (don't miss the u-boat in the basement!). Some days have hands-on activities for kids, check their website for more information. See the surfers in downtown Munich, they're there even in cold weather. Also hit some of the bigger biergartens in which you can get a meal then let your child play on their play equipment outside (outside seating closes during winter).
Rothenberg odT and Nurnberg are similar enough for your short trip you should chose one to go to or skip them and concentrate on Munich, Salzburg and Fuessen/Garmisch.

You didn't state your dates of travel, so just to be aware, Christmas means lots of closures. Most Christmas markets will close by the 22nd or 23rd. On the 24th, the country almost shuts down about 14:00, except for public transportation and hotels. If you want a Christmas Eve dinner, you need to have reservations at a nice hotel that offers that. The 25h and 26th are official holidays with all stores closed. Many museums and other tourist attraction, as well as restuarants, may or may not be open, so you need to check their websites. In large cities, those 2 days can be quite busy in restaurants though. Some places like Zoos, or movie theaters also do a good business on the 25th and 26th. Ski resort places often offer Christmas packages, that will include special dinners, sleigh rides, etc. that would be fun for the kids, but they are often booked up by now. I would give it a try though, for places like Garmisch, or someplace near Salzburg.

Does anyone have recommendations for a nice traditional Christmas Eve service in on of the churches in Munich? Also can you recommend a nice restaurant In Munich to have Christmas Eve dinner?

That is just it. All the restaurants will be closed on Christmas Eve. You need to make arrangements with your hotel or reserve in another hotel dining room.

Thanks all! We will be staying in an apartment so our hotel won't be able to arrange it that way. I can try looking blindly through tripadvisor for big hotels in Munich that may have open restaurants, but if anyone can recommend one that surely would be appreciated!