Regarding connecting devices, a magic 'fix' might be to check the Use Legacy Connection on the Companion before scanning the Qr code. Please try that. What happens?
Make sure you are connecting to App Inventor using Chrome or Firefox. Do not use Edge, it is incompatible with App Inventor. Using Edge seems to work but then, unexpectedly strange things happen...perhaps what you are experiencing?
You said "are there any limitations by the server, if there are too many connections from one IP? The overall internet connection at school was fine, we could use other websites without limitations." App Inventor uses WebRTC, it is not a web site, it is a Web app. Yes, there possibly are too many connections on your network, this will depend somewhat on your school's Internet provider. A 503 Service Unavailable error is an HTTP status code that means the website's server is simply not available right now. Most of the time, it occurs because the server is too busy or because there's maintenance being performed on it. MIT has several servers, when the load becomes heavy, it provides additional servers so MIT is probably providing adequate access. If that were not true, all users would have this issue; that is not the case
Please read the link and the links within the link. There might be something in that discussion that will help you.