Holyfuck after hours of trying out every single possible fix and searching the internet I finally found a solution by myself and fixed the problem and could install steam now.
:D

I always used the steam client from the official steampowered.com page before…

Now I just downloaded the same client (same version etc) from an inofficial page and it suddenly works without any problems. Seems like the client of the official page is currently corrupted or something.

Whatever now it’s time to download and play videogames :P