Paris in 6 Nights

On my second trip to Paris, I visited the Army Museum for the first time. I highly recommend it if you are at all interested in history. We only had time for the WWII wing and Napoleon's Tomb, but it was worth the visit.

In December I'd want to visit the big department store of Galeries Lafayette Haussmann. Bonus is it's right across from the magnificent opera building.

I think Versailles is still worth a visit without the gardens in full swing. There is plenty to see inside the castle and the adjacent buildings.

We visited the Lourve, the D'Orsay and the L'Orange. We were advised to stick to the smaller museums but we went to all of them, but honestly we liked the smaller ones best. Go to the Eiffel Tower early evening/night when it lights up. You will never forget it. Eat at Beau Repaire near the Notre Dame. Get Domaine Abotia wine if you can find it. Best wine I ever tasted and I like beer better usually, though not a big drinker. If you like walking tours, we took one with Sight Seekers Delight. It was a highlight of our trip. Ask for Stephanie. It was all about the Jewish History of Le Marais. Do a little cruise on the water in the evening when you first get there. Do a night bus tour to see the City of Lights at night. Have a blast.

After reading all your comments and advices, I feel like I have more than 6 days worth of places to visit haha. Thanks a lot for the prompt reply. I will also appreciate more suggestions!

You have a busy schedule, but (of course there is a but) weather and opening hours of different sights will change with the seasons and days of the week. A lot of the exact organization may be last minute for that reason. And, for example, the Orsay was going to be open one evening when I was there so that changed the schedule by making it an early dinner and later quiet and uncrowded stroll through Impresssionism.

There are lots of organized tours out of Paris to other sights - Giverny, WWI battlefields, even Normandy. Giverny for the artists in the family, the Invalides (Army Museum) for anyone who likes suits of armor, etc.
http://fondation-monet.com
http://www.musee-armee.fr/accueil.html

Have a great time!

You can include these sights on your itinerary for the 2 open days you listed:

Day 5 - Possible day trip to Fontainebleau Castle

Day 6 - Day trip to either Reims or Chartres

Since you'll be there in December, I would definitely factor in some time to visit the city's beautiful Christmas markets! Here's a good overview: https://www.parisinsidersguide.com/10-best-christmas-markets-paris.html

If you have any interest in champagne, a day trip to Reims is easy and delicious. We visited the cellars of Pommerey and GH Martel and thought it was a good combination of a glitzy tour (Pommerey) and actual champagne tasting (Martel). Definitely buy tickets ahead of time for Pommerey. Reims also has a beautiful cathedral. (Although, be aware that the name of the city is pronounced "Rance" rhymes with "France".)

One other museum I would add is the Orangerie, which is a must if you like Monet. It's small, generally uncrowded, and you can see Monet's paintings in the setting for which they were originally designed. I like it better than the Orsay.

Please carve out 2 hrs for the Luxembourg Gardens. A wonderful park, good place to relax for a minute, and be around other Parisians enjoying themselves vs rushing around.