Even though I was so happy to get an NES from Jeppe and Peter, the timing issues with the PAL NES has bothered me enough to buy a NTSC one to replace it. I bought a refurbished one from Lukie Games including Super Mario Bros. and Duck Hunt, two controllers, and the Zapper. The whole thing was 180 USD (1174 DKK).
I already got my US NES in the mail, only 8 days after ordering. And it works! And as I expected, the Zapper doesn't work on my stupid laggy LCD TV. I'll have to try it on a CRT at some point...
I connected my NES to Jeppe's old TV in my bedroom, and tried my luck at some Duck Hunt. I hadn't played it before, and was surprised that GAME A felt a bit too easy.