Your trip is fairly short and you've identified a lot of good destinations in Denmark. Personally, I wouldn't try to shoehorn Norway into this trip. There's a lot to see in Norway: Oslo, the fjord area (the Nutshell route is just part of it), Bergen, the west coast and more. There are some good outdoor activities in Norway your teenagers would probably enjoy. The Voss area, near the Nutshell route, has a variety of options, I think, and there's good hiking just outside Oslo as well as a wheeled bobsled run in Lillihammer. The Norwegians are very outdoorsy folks.
It takes time to travel around Norway, a country with very challenging geography; you keep bumping into water that has to be crossed, often with nary a bridge in sight, so you wait for a ferry. Most towns have a rail or bus link, but the bus links to small places may be only once or twice a day, so renting a car for part of the trip can be very desirable. Because of those factors, Norway's really not a place to go for just a few days.
In case you are not aware, Norway is also eye-wateringly expensive. On top of that, the tourist infrastructure in rural areas hasn't kept up with the demand, so you'd probably struggle to find lodging for July at this point. Instead, I'd recommend doing some research on Norway; it would be worth reading old threads in this forum going back to a couple of years before the pandemic. If it sounds like a great family trip (which I think it would be), you can dig into the details when you get back from Denmark. The first thing to do, I think, would be to nail down lodging along the Nutshell route if you want to spend one or more nights in that area. It will be a lot harder if you wait until 2024 to look for rooms or apartments.