Copenhagen, Prague and Rhine Valley with adult sons

So you have a maximum of ten days in Europe?

Taking the train from Berlin to Cinque Terre doesn't look practical--something like 17 hours. I would want at least two full days in C.T. on a trip like this and suggest that you think about a good outdoor destination in Germany instead. You'll find castles in Germany, too.

In addition to important traditional art and ethnographic museums, Berlin is especially great for WWII and Cold War sites, so that sounds like a good fit for you.

"Any suggestions for another stop between Berlin and Cinque Terre's if we take the train all the way down? Or should we fly out of Berlin to save time. We enjoy walking/exploring, moderate hiking, not too many museums. Will see a couple of castles, suggestions?"

After Copenhagen you have 7-9 days. It takes 7 hours for Copenhagen-Berlin - most of another day. You'll need at least 4 nights in Berlin (which yields just 3+ days, not really much for this big a place.) So that leaves 3-5 more days. You would spend 17-20 hours of that time on trains to reach Vernazza. Then you'd travel around the Cinque Terre a bit, then travel again to Florence. It's a whole lot of moving around for the short time you have, and you'd be shortchanging your experience in Germany.

I'd suggest instead that you do your walking/hiking/castle-snooping in the Middle Rhine Valley, a UNESCO World Heritage site that is stuffed with attractive half-timbered, old-world towns, castles, and good trails. The train trip to Bingen at the southern end of the Valley takes 5-6 hours.. This compact area can be explored over 3-4 days - then fly out of FRA airport outside Frankfurt (about 1 hour from Bingen.)

http://whc.unesco.org/en/list/1066
http://www.welterbe-mittelrheintal.de/index.php?id=288&L=3
Bacharach: https://upload.wikimedia.org/wikipedia/commons/4/4c/Bacharach_BW_9.JPG
Marksburg in Braubach: http://www.marksburg.de/en/
Rheinfels in St. Goar: https://www.youtube.com/watch?v=qxFF80wORNQ
Burg Eltz (near the Rhine): http://de.best-wallpaper.net/wallpaper/1920x1200/1206/Castles-in-Germany-Burg-Eltz_1920x1200.jpg
Braubach: http://djtravel.homestead.com/files/braubach_-_hotel_1610.jpg
Rheinsteig Trail: http://www.romantic-germany.info/Rheinsteig-Trail.4134.0.html
Rhine Castle Trail: http://www.romantic-germany.info/RheinBurgenWeg.6751.0.html

I agree with the previous posters, with only 10-12 days I think you should skip CT and focus on Denmark and Germany. The train from Copenhagen to Berlin is quite slow, since it also involves a ferry crossing (which some people find very fascinating for some reason, since the train rolls onboad the ferry). Flying is faster and sometimes also cheaper, I'm heading to Berlin next weekend and I'm flying.
For castles i second Russ' suggestion of the Middle Rhine Valley, it's castles gallore, however only Marksburg and Burg Eltz are 'real' historic castles. The rest are either ruins or have been heavily rebuild in the 19th and 20th centuries. They are spectacular nonetheless.

For castles and palaces in Denmark try the popular day trips from Copenhagen to Frederiksborg Castle in Hillerød, a spectacular rennaisance palace. https://en.wikipedia.org/wiki/Frederiksborg_Castle
Or Kronborg Castle in Helsingør, famous as the setting of Shakespeares Hamlet https://en.wikipedia.org/wiki/Kronborg
The town of Helsingør is also very pleasant.
You could also visit the town of Roskilde and see the Cathedral and the Viking Ship Museeum.
All places are reachable by train from Copenhagen.

I think that only 4 nights/ 3 days in Berlin is short changing one of the most interesting cities in Europe. CT is too busy and overrated IMO.

Flying from Copenhagen to Berlin takes about an 1hr 30min vs. 8hrs on the train. It's also cheaper. The airport in Copenhagen is well connected to trains and the Copenhagen Metro, so it's super easy to get to.

Same thing with Berlin to Cinque Terre, cheaper and faster to fly.

Agreed that there's plenty of variety and opportunity up north, without traveling half as fare as Italy. But if you need a direct flight between Berlin and Pisa, RyanAir has one.

Thank you everyone. I guess we need to rethink going to CT. I removed it from the list before, but I added it back as it's #1 on 'my wish' list (and I'm making all the plans :)! If we go to CT, given our timeframe, flying from Berlin makes much better sense. We wanted to enjoy the trains in Europe but I suppose taking the train from Copenhagen to Berlin will be enough. Sounds like we should spend a little more time in Berlin. Russ, thanks for the Middle Rhine Valley suggestion. Looks like a lovely area and flying out of Frankfurt might be a plus. Thanks again!

Covering from Copenhagen to CT in a 10 days (2 are travel) is too much so I'll focus my recommendation on Germany.

First I'd suggest a flight to Germany (based on your itinerary). When I was in Copenhagen, I considered taking a train to Hamburg to visit a relative, both the travel time and expense were prohibitive. I also considered a rental car option between the two with the same result.

Fly to Germany, then use the train to get around.

I find Berlin is somewhat isolated from many nice places in Germany. Travel times to Berlin make it out of the way and expensive in terms of time invested. If it's your top must see, however, by all means include it in your itinerary.

Both Copenhagen and Berlin are cities with lots of activity choices. I would want at least three full days in Copenhagen and more in Berlin. That doesn't leave much time to see more. A train to Dresden for a day might be a good side trip. Prague isn't far by train but giving all three cities the time they deserve may be pushing it.

From Copenhagen, the day trips I'd recommend are Roskilde (Cathedral and Viking Ship museum) and/or Fredericksborg.

Did the OP edit their original post? Nowhere does it now mention Berlin or CT? It's now Copenhagen, Prague, and the Rhein Valley.

Yes, I adjusted our trip from CPH, Berlin and CT to CPH, Prague and Rhine Valley.

Fly from CPH to Prague. It is too far for train. From Prague to Rhine Valley you can either fly to Frankfurt then by train to Mainz or go by Deutsche Bahn run bus from Prague to Nuremberg then by train to Mainz with one change in Frankfurt. The fastest connection (besides flying) appears to be this one:
Bus (in front of Prague Main Train Station) leaves at 9:40, arrives in Nuremberg (in front of Main Train Station) at 13:18.
Train leaves Nuremberg at 13:29 arrives in Frankfurt at 15:36.
Train (direction Hamburg-Altona) leaves Frankfurt at 15:44 arrives in Mainz at 16:18.
All the way from Prague to Mainz takes about 6 and half hour. If you buy nonrefundable ticket ahead enough it is 39 Euro per person. Otherwise 102.10 Euro.
The most scenic part of Rhein is between Mainz and Koblenz. You can go by boat to St.Goar or Bacharach, stay overnight, then continue to Koblenz. If you have time visit Mosel River valley. Nice scenery. Many castles around Rhine and Mosel. Lot of wineries with an excellent white wine.