I have a laptop that is a bit slow and no one will use it as it is because it is very slow because of the Windows updates. It is such a mini laptop and has no LAN. USB TO LAN would be an option but for 10 euro I could rent a root server with more RAM. Running it over VM is bad because Windows uses way too much. Could I still use Wi-Fi somehow?
Your question is completely unclear. Of course you can connect a computer with Debian as the operating system to the network via WLAN. But what does this have to do with a root server? Do you want to use the laptop as a server? Sure, you can do it. Whether a Wi-Fi connection is sufficient depends on how good the Wi-Fi connection is and what you plan to do with the server.
Your question is very confusing. Maybe you can correct me if I misunderstood something or ask your question again with orderly thoughts.
What I understand is that you want to know if you can install Debian on your mini laptop and set up a network connection with Wi-Fi?
The answer would be:
Basically yes, as long as the WLAN card in the laptop is not too exotic. You can also test this very easily by downloading a Debian image and creating a bootable USB stick with it. Then you can test Linux from the USB stick and check for yourself whether everything works before you flatten the hard drive and install the new operating system.