Sindre: Now you have too many ideas. At this point, here's what you should do. Step back for a couple of days and look really hard at Tokyo or Seoul, as to whether they fit in your budget for flight and for on-the-ground expenses. My guess is that with both flights, hotel, food, local transportation for about 4-5 day trip, even with the air-hotel specials you found, you will be spending a total of USD 1200-1400 for this short time--lower end for Korea, upper end for Japan. Unless you have a very strong desire to see either of these places right now, I suggest that your time and money is best spent elsewhere.

North Korea: If this is a STRONG interest, then you must go with a tour and they will take care of the NK visa, as you cannot do that on your own. It all is done as a package. Most foreigners go with a Beijing-based agency such as Koryo tours (google for their website). These will not be cheap, likely almost as much as going to Tokyo or Seoul options. There are somewhat cheaper tours organized for Chinese visitors but without being a Mandarin speaker, it will not be a good choice for you. Again, if you don't have a strong burning desire to see North Korea, then skip this.

China: You seem to have discounted China on the basis of visa/invitation formalities and I think yes, you are worrying too much. There are plenty of Norwegians running around touring China and after the initial paperwork hassle to get the visa, they get inside and then do what they want. There is no shortage of places to see in China, much more cost- and time-effectively for a few days.

If you decide to go international, then this thread stays here. And obviously, you need a double-entry Chinese visa. If you decide to stick with the entire trip inside China, a single-entry visa will do, and then we'll move discussion over to the China forum, where the experts hang out, and help you craft your extra days, based on your personal interests. And help you strategize how to deal with Chinese officialdom requirements with minimal trouble to you.