Nice has become one of my favorite places on Earth, and I was just there for a week, 2 weeks ago. We also were there for a week in early August, when it was very hot. In mid-to-late November, it was much more comfortable, and I had a light jacket. A few shops were decorating windows for Christmas, but I've not been in Southern France at Christmastime, so I can't tell you exactly what to expect.
I've also not traveled with children, so I don't have a lot of specific recommendations, but the park at the top, overlooking the old Nice and the sea, has lots of playground equipment on which energetic kids can climb and play. There's usually an elevator you can ride up to the park/overlook, but it was down for repairs just now. It should hopefully be up and running by next November, but we simply climbed up from the old town, so it's still accessible. A car could be a hindrance in Nice, but the tram and bus systems work very well. A great side-trip would be Antibes just down the coast, with a fun, outstanding Picasso museum. If your family is like us, everyone would especially enjoy the room with all the plates on the wall, many with faces in the center.
We drove from Saint Remy this summer up to Lyon, then down to Nice. A farmers' strike closed the Autoroute one day and strangled traffic for hours. Hopefully that wouldn't happen in November/December! Lyon, a big city, is another place to park a car and leave it while you ride the metro, bus, or a taxi. It was impressive for our short visit this summer, and this past weekend I just booked an apartment for 3 weeks in the 6th arrondissement to have more time in the Lyon area and get to know it better next June/July (slow travel is good travel).
We may head up to Alsace in the summer after Lyon, and I read thru Rick's chapter in his France book yesterday. He suggests a car is helpful if you're planning on touring the wine route. If we go, we'll probably look for a place to base ourselves in Kaysersberg or Eguisheim, just outside of Colmar. I'd bet the Christmas atmosphere in Alsace would be something special.
Even for someone used to driving on the right, the roads in parts of southern France (around Arles, for example) are challenging, with speeding, tailgaiting drivers and somewhat narrow lanes.
