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Should I imagine it like a TCP/UDP port (per-machine rather than per-user)? Can I connect to an other users DISPLAY? Is it protected somehow?

Can I list the currently used DISPLAY numbers for one user? Is it possible to find one free DISPLAY number that I can still use?

Where could I find out more about these?

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Should I imagine it like a TCP/UDP port (per-machine rather than per-user)?

It actually is a unix domain socket for local users, and a TCP port (if enabled, on modern X servers it's disabled by default).

Can I connect to an other users DISPLAY?

Yes, with proper authorization. See xauth and xhost.

Can I list the currently used DISPLAY numbers for one user?

Display numbers are per X server, and not per user.

Is it possible to find one free DISPLAY number that I can still use?

ps axu | grep Xorg should list all X servers, you can see which display number they use. Or look at /tmp/.X11-unix/ to see the unix domain sockets. Possibly there are variations for this among distros.

In general you should have an idea how many X servers are running on your system, if you have root rights and configured it ...

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