All that background and your priorities were helpful. Here's how I think I'd handle it.
Days 1-4 in Berlin (3 nights)
Day 4: some sightseeing... then leave in the mid-late afternoon on one of the DIRECT trains (4 - 4.5 hrs.) to Cologne Hbf (main station.) 1 night near main station, 4-4.5 hour train ride. Cologne Hbf is a convenient location for rooms and meals.
Days 5&6 in Cologne, Cochem, Burg Eltz, Wierschem (2 nights in Cochem)
Day 5: morning visit to the Cologne Cathedral and/or Nazi Documentation Center, a memorial to victims of the Nazis, not far from the station. (This is a solid alternative to the concentration camp, which isn't recommended for children anyway and would take up a large portion of your day.) Lunch. Early afternoon train to Cochem (12:53 - 14:41, change in Koblenz) -or- catch the DIRECT train, 14:17 - 15:54.) Drop bags and enjoy the town. If you hustle (shuttle, taxi?) there's a 16:30 tour of Reichsburg Castle if you're interested. Cochem feels very old-world, not unlike Rothenburg. Stay in the old-town part and have a good wander there.
Day 6: You are in luck. Wierschem is on the route to Eltz. Morning train from Cochem to Hatzenport (20 min.) where you catch Bus 653 (Castle Bus) which will drop you at Eltz Castle's parking lot after about 25 minutes. The bus stops in Wierschem (Hinter der Hecke bus stop) a few minutes before reaching the Eltz lor. I suggest you visit Eltz first and see Wierschem on the way back. Then return to Cochem as you came (catch an afternoon river cruise?) There are lots of train and buses for this trip all day long from Mon - Sat, though on Sundays there are fewer buses. See bus schedule to Wierschem and Eltz below:
Day 7 - Marksburg Castle, train to the German Alps (1 night in Garmisch-Partenkirchen)
Leave Cochem early on a morning train to Koblenz. Drop bags in locker, train to Braubach (10 minutes) on the Rhine River for a tour Marksburg, outstanding example of a true medieval Castle. Train back to Koblenz for bags. Then you have a big train trip. Koblenz > Garmisch-Partenkirchen (13:48 - 19:54?) I would stay at the Fraundorfer and catch the Bavarian evening fun there.
Day 8: public transport to Innsbruck (as planned) is easy from G-P.
You should book 3 train trips well in advance...
- Berlin > Cologne
- Cologne > Cochem
- Cochem > G-P with stopover in Koblenz
Prices right now for July/August look very cheap now but will be hugely expensive if you wait. Kids under 15 travel free with parents.