Apple - Does XQuartz work on macOS Mojave?

Yes, the latest version of XQuartz works just fine on my macOS Mojave machine with absolutely zero issues.

Yes, but one limitation is that it does not support Retina displays, meaning that the UI is quite pixelated on an Apple Retina display.

To add a bit to @abc's answer:

1. I've had good luck with XQuartz, but looking at the XQuartz GitHub page, it would seem there's not a great deal of current activity (maintenance or new development) taking place.

2. I only use XQuartz over an SSH connection: ssh -Y user@hostname

3. You will need a ~/.Xauthority file on your client machine (not the Mac). It should be generated automatically, but check for it.

4. I've got a few notes that might help you on my GitHub page.

Let us know if you have more questions.