It turned out to be a series of small mistakes on my behalf. My solution is posted here, but I'll repost it here as well.
This turned out to be a series of inaccuracies on my part. I'll post all the stuff I ran into in case anyone else has similar issues. I think this is about what happened:
1. Everything was set up correct initially, but I changed www.mysite.no to localhost/var/www instead of just localhost.
2. Then I changed to a fresh installation of WordPress with a working (empty) database. The real database did not work (as would be expected).
3. I then updated the real database by switching localhost/var/www to just localhost. This did not work, however. At this point I had probably messed up something else. Also, it seems that the proper solution is to use the IP address of the server, at least in this case where I use the browser in another computer on the network.
4. Then, I created a new test database with all the data and changed www.mysite.no to 192.168.1.2, which is the IP address of my local server, and copied the WordPress files from the live server. This made the site work as expected. (As a side note I ran into some issues with "Failed to open stream: Permission denied". This was fixed by running sudo chmod -R 755 /var/www.)
5. Now I had some trouble with permalinks (which is an entirely different issue). The .htaccess file was missing, so I copied it from the live server and re-saved the permalink settings at Admin --> Settings --> Permalinks. Then I switched AllowOverride None to AllowOverride All in /etc/apache2/sites-enabled/000-default and ran sudo service apache2 restart.
So this was a series of mistakes on my behalf. I am still a bit baffled that WordPress does not provide any error messages when the error is in the database URLs, however. Hope this can be of use to others.