"Do you think we should take a train to Luxembourg City and drive to Cologne then to the Mosel/Khoblenz area."
Cologne is not a good base for exploring/visiting the Luxembourg/Germany border zone, the Mosel Valley or the best part of the Rhine (south of Koblenz.) Travel4fun appears to think Cologne is a peculiar choice - and I do too. It's hard to say what you should do without understanding what your purpose in Cologne is, but if you aren't sure of your goals there, my first instinct would be to drop it. A Paris > Luxembourg > Mosel Valley > Middle Rhine Valley > Munich routing makes better logistical/geographical sense and would probably be a more enjoyable trip. Trier, Burg Eltz, Cochem, and the old-world Rhine towns of Boppard, St. Goar, Oberwesel and Bacharach (huddled together south of Koblenz along the west bank of the Rhine) are good suggestions along this route.
Also, I don't quite understand the reasoning behind a switch from the train to a car to the train - and then to a car again for Munich - Garmisch. Perhaps you need a car when you "...drive to the Luxembourg to research family history" as smaller towns in the countryside of Luxembourg might necessitate a car. The city of Luxembourg itself probably does NOT require a car and is easily reached by train. The same is true for Trier, all the Rhine towns, Munich and Garmisch. Of course, with 7 people, you wouldn't need just a car but an oversized vehicle of some sort, or two cars. There are direct trains between Luxembourg (or Trier) and the other Mosel towns (Cochem, Koblenz and others) along the Trier-Koblenz train line, which follows the river much of the way and offers great scenery as well.
Cochem
Bullay
Treis-Karden
Driving the entire Mosel River is very scenic as well, but because it is incredibly windy/curvy, it can take next to forever to cover all of it, and your time is quite short.
As you know, there's no train to Burg Eltz, but if you don't wish to do the walk, there is a weekend bus - the Burgenbus - from the Mosel town of Hatzenport.
The Rhine south of Koblenz is also very well served by train. The stations are all close in and the towns are all small and close together - only a few minutes between each. It takes less than 20 minutes to cover Boppard - St. Goar - Oberwesel - Bacharach by train. The scenery is stunning here as well and the tracks are right along the river. A river day cruise is another way to enjoy this area.
Day passes make travel in the Rhine/Mosel region simple and inexpensive.
And there are direct trains between Munich and Garmisch that take less than 1.5 hours; similar day passes work for this route/area as well.
