Hamburg and Berlin help and anywhere else?

Berlin and Hamburg have a lot things to offer and explore what matches to your interests. In both cities a car is not meaningful for a tourist. And yes, Berlin changed a lot, and in some corners it did not really. Meeting local people is easy on week and flea markets and other events. Berlin needs minimum 3 full days, plus 1 day for Potsdam.

Be aware that train travel time between Berlin and Hamburg will take longer from Aug 1st on (see this thread).

Denmark has a more relaxed and pragmatic culture; Copenhagen has also a slightly different one - what I heard from a Dane living in Copenhagen. People there like enjoying summer outdoors at local events or beaches. Ribe, Odense or Aarhus can be worthwile stopovers.

"Would rent a car if there are smaller places to visit."

A car is not needed to visit smaller places in Germany.

Please have a look into Lüneburg (Lueneburg), Lübeck Luebeck) and Schwerin. These three are well known to Germans but not so well to North Americans. I doubt Rick has ever had these places on his lips or penned them in his materials anywhere, though you may find a few comments about these places on his site by doing a forum search. You'll need to consult other travel resources whose writers have been to these places.

This link has solid information for a visit to Lüneburg.

I would suggest taking the train from Hamburg Hbf to Schwerin, much easier than from Berlin, to see that famous and lovely Schloss.

Are you interested in going north of Hamburg, ie into Schleswig-Holstein, or to the west in the direction of the North Sea towards Oldenburg? Some very interesting towns in that region eg, Schleswig and Eutin/Holstein along with their historical museums.

Bottom line: Schleswig-Holstein is well worth your time and energy spent visiting and exploring these smaller historical and cultural towns.

You could consider getting to Berlin and Hamburg before the closures on August 1, then take the train to Copenhagen and fly home from there.

I liked Hamburg and you would have to stop there anyway if you are taking the train to Copenhagen from Berlin. So with 12 days, you could easily spend 5 days in Berlin, 4-5 in Copenhagen and 2-3 in Hamburg.

I could write (actually, I have written) TONS on the amazing things to see and do in Hamburg.

Based on your interests, I would also recommend day trips to Bremen, Schwerin, Lüneburg, Lübeck, St. Peter-Ording, Helgoland, and my personal favourite, Glückstadt. All of these are SUPER easy to reach by train from Hamburg (except of course Helgoland, which you reach by catamaran ferry from Hamburg).

Helgoland is walking, nature, birds, red cliffs, views, water, and wind. It's also a bit of a cultural experience as the islanders have their own subculture, history, and even language.

Glückstadt is history and small town architecture, local Matjes (food), dikes and sheep, farmers markets on Tuesday and Friday, and a totally small town feel, and a truly special harbor that is essentially a smaller and not crowded version of Copenhagen's Nyhavn. It's often overlooked, but it is really a hidden gem. Ambitious bikers can bike 45km along the dike to Glückstadt, then take the train back to Hamburg.

Lüneburg, Schwerin, Lübeck, and Bremen are all great day trips for pretty much anyone--all four are beautiful, architecturally and historically interesting.

St. Peter-Ording has probably the most interesting and one of the prettiest beaches you will ever see.