(I’m the dev of Alexandrite) Officially there isn’t yet a way to self-host it without modifying the code (unless you host it on one of a few specific cloud providers like the official Alexandrite site does), but now that people are self hosting anyway I think adding support for it is going to be higher up on my priority list!
Yes, launching the game launches Ubisoft’s launcher and you have to sign into that to play.
Also if you use Proton it doesn’t auto close the launcher when you quit the game so it’ll run up your play time while the launcher is minimized to the system tray.