Also das DNT von Norwegen hat mir folgendes geantwortet auf die Hundefrage:
As long as your dog is leashed, you can bring your dog anywhere you want in the nature reserves surrounding E1. You do not need any permissions. There are only a very few special bird reserves around the coast where human (and dog) activity is restricted.
Dogs are allowed in some shelters, and probably not allowed in some. You should check the website ut.no (its only in Norwegian so you have to use google translate), where you can see detailed information about all DNT cabins there. See one example here. Under “facilities”, a gray field below the text, you can see if its allowed to bring a dog if the symbol of a dog is present. In the example I used, it is allowed to bring a dog.
You will use many shelters run by the DNT. The DNT is divided into several local associations, and it is up to each local group whether dogs are allowed at the shelters or not.
Also in Norwegen, mal keine Probleme. Antwort Schweden steht noch aus.
Egal wie weit der Weg ist, man muss den ersten Schritt tun.