The buyer nationality doesn't need to be Danish. Anyone could buy your item on Amazon and receive it, then ship it to you to the address you want. Or your order and pay the item and give the address of the person. Obviously, it would need to be someone in the US. Of the person doesn't want to give his personal address he can pick the box at a "Pick Up Point"
I found another solution and apparently everyone is using Amazon.de and Amazon.co.uk which ships to Denmark. But just to be sure I got you right. If I find someone who could buy the item from a danish website, all he has to have is a credit card? He doesn't have to be a Danish citizen or anything and he could just fill the shipping info with my details?
Yes, he just signs up, fills his/her information, orders the item and pays it and that's all. And yes he could also fill the details with your address directly.
Usually, they ask you an address each time you place an order, in case you want to receive the item in another address that you provided before. You work, parent address, ...