OK, your itinerary isn't as bad as I feared.
Much of Germany shuts down during the Christmas holiday, but a few things start slowly opening up after the 1st, and the 6th is the official "end" of Christmas. The smaller the town, the more dead it will be.
So I wouldn't recommend staying in a Black Forest town on the 31st as it will be dead except for kids shooting off fireworks.
Colmar or Strasbourg are fine choices, they don't shut down as thoroughly as Germany does after Christmas. Both have Christmas markets that run through at least the 31st, although Strasbourg shut theirs early a few years ago due to terrorism threats. So things can change. Most people here prefer Colmar because it's smaller and more charming. I like Strasbourg because there's more to do, but both are lovely and you will be happy in either and should absolutely take a day trip to visit whichever one you don't stay at.
I personally don't think driving from Alsace to Fussen in a day is that long, but I would want to be somewhere "big" for New Year's Eve. I'm a little biased but I think seeing Germans go crazy with bottle rockets on New Year's Eve is a cultural experience worth having, and thus would suggest Stuttgart as your overnight on the 31st. If you left France early enough you could have time to meander through some the more scenic Black Forest towns (I like Alpirsbach for their brewery/monestary and Calw for the half-timbered houses) and get to Stuttgart by nightfall which is when people will start setting off fireworks. Find a hotel with a view of the city and enjoy the show, then go to Fuessen the next morning (2 1/2 hour drive or so).
That said even in Stuttgart, ALL the museums and attractions here are closed on the 31st and the 1st so it would only be for NYE. But at least restaurants will be open, some of them, anyway (reserve!) and the Wintertraum ice skating rink is open until 4pm. In a tiny village you'll be lucky if anyone but the Doner Kepab shop and a smokey dive bar are open on the evening of the 31st.
Apparently Strasbourg and Colmar also go crazy like Germany on NYE with people setting off random fireworks, so you could just stay in one of those places and make the longer drive on the 1st.
The rest of your trip should be OK because those places are touristy enough that they don't shut down completely the way much of the country does. But be aware that even Neuschwanstein is closed on the 1st. Double check that the things you want to do will be operating on the days you plan to do them. You may need a slightly more structured trip than usual in order to not be frustrated by closures.
