I have a root server and someone asked me if I could give him a Minecraft server, he pays for it too. Would it be legally allowed to "rent" the server without having to register anything commercially?
No. You can only rent something commercially with a registered business.
You can "share" the costs. That would be legitimate.
But I'm not a lawyer;)
Dannyotti, does that mean I can say joa give me some of the costs that "we" need and I host the server?
That's fine. You are not allowed to make a profit and not declare it as "sublet". Something like that is commercial.
You can also get together for a computer game and share it. Why shouldn't it be any different with a server?
Just how much you ask of him is up to you. But something like that must not happen all the time, then it is no longer a "friendship service".