You're at risk of spending a huge percentage of your vacation time in transit from place to place. It's a temptation everyone faces because there are so many wonderful options. It is, however, not the way to have a wonderful trip, especially if there are children involved. I think you may perhaps not have researched travel times. You propose 1-1/2 days in Lucerne, but even if you are willing to take a 7:57 AM train from Paris, you will not reach Lucerne until 1:55 PM. It will probably be 3 PM by the time you are settled into your hotel room and ready to head out to see the city, so that's not a half-day of sightseeing. And are you really going to try to get your family to the train station in Paris early enough to catch that 7:57 AM train?
I find the Deutsche Bahn website the easiest to use when looking for train schedules. You can easily check the travel times between places you hope to visit. Pick a date at least two days in the future so you can see all the trains and set the departure time to whatever morning time you think might work for you. For final planning you'll need to use precise travel dates, but quick-and-dirty will probably give you relatively accurate durations.
To get prices and buy tickets, you'll need to go elsewhere unless you decide to travel to Germany.
I'm concerned about the time you propose to spend in Paris, which it seems is a must for you. In giving yourself two nights there, you are probably limiting yourself to one day of sightseeing (Day 2 in Europe) and a partial day of wandering around sleep-deprived and jetlagged. That certainly would not be enough for me on my first trip (or any trip) to Paris.
I suggest you have a serious conversation with your travel group about the impossibility of doing what everyone wants to do in only ten days (which is really only 9 non-jetlagged days, if you're lucky). You'll all have a lot more fun if you travel less and really see two or at most three places that are well-connected, ideally by rail.
