I'm not in Copenhagen (currently), and I'm not aware of a shop per say other than what Henrik pointed out.
I do know that Fritz Hansen -
http://www.fritzhansen.com/en/fritz-hansen/contact/HQ-and-Management is just north Copenhagen, but not heard about it being possible to see manufacturing.
But if you're visiting and are looking for furniture stores then you can head up Store Kongensgade. Its close to the Marble Church and the Royal Palace. You can find NytiBo there.
http://www.nytibo.dk/
http://www.montana.dk/home/
The latter one is on Bredgade.
Illums Bolighus also have a variety of european furniture.
Paustian is further away, reasonable accessible by the s-tog (t train).
http://en.paustian.com/index.jsp
Henrik may have other ideas.
If you check out that 'section' you'll also be close to Nyhavn where you'll find the smallest house 'width wise' "Nyhavn 4" as well as a lot of history. The house is just a sliver really. You can touch both sides of the walls no problem.
Don't miss a visit to a Danish Pastry store like Reinh van Hauen -
http://www.rvh.dk/historie.html Other exists as well, but that one was the one I could find. Get a 'real Danish'.
There is one on Store Kongensgade 45
The Brewery next to Tivoli is a good place, at least last time I was in town.
In Iceland try the Blue Lagoon. Take one of the tour busses to see the nature. It can be on the expensive side, in comparisone to the us, just so you know.
If in Stockholm do take a walk through Gamla Stan as well.
