Recommendation for a shuttle service and tickets to the Nutcracker

Shuttles in Paris have a reputation for unreliability and for four people are likely to be more expensive than a cab. The obvious best bet is to get a cab in the official queue at the airport. The dispatcher will match you with a maxi cab with plenty of room for you and your stuff. You pay 3 Euro for the 4th person and one Euro for each piece of luggage in the trunk. Expect to pay between 50 and 60 Euro for the trip. Paris is one place where a cab is always the best way to get from the airport to the hotel and back when you are a group. The train would cost you about 40 Euro.

Albert, generally a Taxi is more reliable than a shuttle service, and you can likely easily find a taxi at the airport large enough for your needs.
Taxi's G7 is a solid company in Paris and they have an English phone line you can call: +33 (0)1 41 27 66 99

The ballet sounds great but I'll leave it for others to answer; have a great trip.

I've never been to the ballet in Paris myself, but if you mean the one at the main Paris opera, I just looked on the official website and several days in December are already sold out. Go to https://www.operadeparis.fr/en/saison-2014-2015/ballet/casse-noisette-rudolf-noureev NOW and book your ticket.

If it isn't that one, still book in advance. Any popular theatre/dance performance in Europe nowadays tends to sell out early. Also, book at the official site and be very wary of other places offering tickets for sale online - the bigger demand the more likely it is that unofficial ticket sellers will be scam artists.

If it's the ballet at the Opera of Paris, tickets sell out within hours. If you see some, grab them.

We always go to the opera when in Paris; both the Bastille and Garnier houses are managed by the same on line operation and tickets go on sale several months in advance. You have to be registered and have a number to get in line for purchase on the site. I always get up at 2 am and am ready to phone the moment the box office opens in Paris; even then usually the best inexpensive seats are sold out by the time I get to the head of the line. It is typical for the tickets to be sold out within hours of going on sale. Sometimes for less popular performances some of the more expensive tickets remain available, but for Nutcracker at Christmas I would assume tickets sold out the day they came on line several months ago. That leaves scam artists, re-sellers and checking with the box office to see if anything opens up. Sometimes blocks of tickets get returned.

I would have an alternative plan for something to do.

Note also that your life will be happier if you take less luggage. I learned this the hard way, but sinceParis and London are in the same weather zone, one small suitcase should be plenty if you pack light for travel.

In general, theaters may also release unused or returned seats closer to the performance time, so check again later or at the box office if it's a priority to you. See also this part of their web site: https://www.operadeparis.fr/en/l-opera-et-vous/spectateurs-individuels-particuliers/comment-reserver.

When will you be in each city and for how long? Like janettravels44 said, don't take so much stuff. Whatever you take, there will still be lots of schlepping it around even if you take taxis to and from your lodgings and to and from your modes of transportation.

Each person needs to be able to handle their own luggage while walking -- up and down stairs, long distances both outside and inside airports, in the metro, the Tube, train stations and probably some other places you might not expect.

We carry on 22" bags, pack for a week and a day and travel for 4 weeks+. The answer is layering, accessories, coordination and realizing that nobody's going to notice what you wear and you may not need to dress up as much as you think. AND doing laundry once a week.