And I solved the problem
And it's kinda one of those things when you found the problem to be something you didn't think at all
Anyway, there was one other thing, that was different with my machine, when I earlier tested the Remote Play...
And it was the graphics card... I needed the card to test one other computer and started using the built in GPU. Well, I also changed the display now and was forced to use VGA-connector. So now I decided to use HDMI instead, but the built-in GPU doesn't have HDMI. So I reinstalled the other card. And now that Remote Connection works xD
Oh, boy... Well, now I may buy an USB3-card to this machine, to see if it brings even more speed to my wireless connection...