Car travel from Paris to Vienna then to Nice

The first thing you ought to consider is that you could be encountering really bad weather and closed roads and mountain passes and/or endless delays because of traffic. The days will be really short that time of year. If you have to be in Nice for a return flight on January third, I would seriously rethink the itinerary. Also, I'd plan an alternate route, just in case. Are you leaving Paris on the 22nd of December? I'd spend Christmas in Munich or Salzburg, remembering that on the 24th everything shuts down early afternoon and that the 25th and 26th are both holidays, with stores closed but restaurants and entertainment venues open.

Anna's advice is good. You have 13 days which is quite a lot even as you realize that Christmas Eve, Christmas Day, what the British call Boxing Day, and New Years Day are unlikely to be good travel days. Also realize that this is when loads of Europeans take family trips so it is likely that the roads will be quite busy, and you would be wise to leave plenty of contingency time for bad weather. Without reservations it may be difficult to find dinner and lunch on those days. The three points of your triangle are quite spread apart. Paris to Vienna is 12 to 13 hundred km, and will take at least 12 to 13 hours of non-stop driving. Vienna to Nice via Salzburg, via northern Italy, is a few km less, maybe an hour less driving. If you include Switzerland it will take much longer. Many passes in the Alps are closed in winter so check your route carefully. If driving through Germany check if you car needs winter tyres (tires) fitted. Make sure you have appropriate winter emergency tools, blanket, lights, water and food in case you are stranded. When there are accidents the motorways are often closed for periods of time. You cannot count on being turned around to get off, you will probably have to stop where you are until the road is reopened. The road may also be closed due to ice or snow. Winter was very early last year throughout Europe. I don't want to scare you with these stories - I want you to be prepared. Probably nothing will happen but you want to survive and be comfortable if it does. Good luck with your planning and have a great trip. I'm sure the children will remember it for years. I have not done that exact triangle but have driven all of the triangle over the years, just never in winter.

Nigel raised a good point - winter tires are required in Germany and if you drive through Switzerland you need a sticker for your windshield. I don't know where you get it, perhaps at the border.

For motorway travel in Switzerland a CHF40 Vignette is required. It is good from December of the previous year to January 13 months later - there are no shorter ones. It is very hard to avoid motorway travel in Switzerland, many two lane roads are designated such. For travel on motorways in Austria a Vignette is also required. €7.90 for 10 continuous days, more for a month.
The Austrian Vignette does not cover the toll tunnels or toll roads. The main motorway from east to west in Austria is >>full<< of long expensive toll tunnels. If you travel across northern Italy you will be paying tolls. Same in France. Monaco has no tolls. Make sure that your car rental company and insurance allows you to take the car into Italy if you are going that way. You should get an IDP for all drivers.

Nigel's 12-13 hours Paris to Vienna is on the autoroute/autobahn/freeway, or whatever they are called. It's a loooong trip. We used to make that trip twice a year and found the easiest was on the overnight train, in a sleeping compartment. It was much easier than driving. Nowadays, Air Berlin has very cheap flights, for as low as 20euros each way if you book early enough.

Paris to Vienna at that time of year will be really, really long. My skin crawls at just the thought of driving it with my kids. Just for reference, We went through Munich, down the E55/A10 through Salzburg through Villach a few weeks ago, a 5 hr drive for us on Google. Took 7 hrs. driving. Snowing on the passes already, blowing snow made visibility poor, one way traffic through a 8km tunnel (had to wait at each end to go through), etc. Plus our two 15 min food/restroom stops, totaled 7.5 hrs. Between stops, construction, possible weather, adds more time than you realize to the googlemaps estimates. If you're stopping somewhere overnight, then it's more reasonable, but flying or train is really your best option. A great stop half way between Vienna and Nice are the beaches of Bibione, Caorle, Jesolo area. A bit cold in January, though.

It is so easy to take the train to all those places, don't bother with a car. Be sure to make a reservation, costs a little more but worth it. The earlier you book train tickets the cheaper they are.

Once you are planning a car trip, I'll try to give advice on driving a car there instead of doomsday-ing you about a far, far away land where people hibernate and don't get out of their houses to drive, only to ride trains (j/k). While the high mountain passes are closed for winter, it is still possible to have a wonderful scenery around. Just need time and good patience to deal with the sporadic delay/road closure. I'll suggest scenic routes for you that are open year-round. They take 3 or 2 days each, as the cities are relatively far apart. Paris-Wien: drive to Gevene via Dijon (A39), then go to Lausanne, Bern and sleep in Interlaken or around that route, on a "valley" hotel - not one in a mountain road. The next day, take your car and go via Luzern, Zug, take A3 to Sargans, cross Liechtenstein and then drive the stunning road to Innsbruck, where you should sleep. The next day, go straight to Wien: if weather is good, you can go via Germany (Rosenheim) without winter tires, if not, take an inland (staying in Austria) detour in Worg, re-joining A10 in Bischofshofën and then to Wien.. Paris-Wien (alternative): if you want a route that takes only 2 days, make sure your car is fit with winter tires and go via Germany (Saarbrücken, Manheim, Nürnberg). Wien-Nice: on the first day, drive yourself to Italy via Graz, Villach, Udine, Padova and finally sleeps somewhere around Parma. It's an all-highway drive. The next day, from Parma take A15 to La Spezia, and then you can drive along the coastal highway all the way west to Nice. But stop a couple times if you have time. There are countless interesting spots to stop, you can just choose anyone. The schedules I suggest you doesn't involve non-highway driving outside daylight time, assuming you leave whatever place early.