Hi there,
Regarding your issues:
1) What you describe sounds like the first time scanning process, which takes some time. It happens in the background, so there's no visual reference (maybe that should be improved), but after it's done you should be able to see all content by clicking on the Subsonic icon. When you click on a subfolder, it triggers a "refresh" of that folder's content, so it will show that (if it's already fetched from the server). The Artist folder contains the least amount of entries (relevant to the main Subsonic node or the Music folder below that), so if those are fetched from the server they will show up when you click on it. If you wait a bit more, clicking on the main Subsonic node should show all tracks eventually.
Also, this first-time scan only happens once and the results are cached locally, so you won't have to go through the same slow process again. If you cancel the process however, the cache may not be complete and it will be recreated the next time.
In other words, let it do a full scan first and test it again afterwards.
2) Does it work if you use the same URL on your browser from the same computer MusicBee runs? I have a custom *.subsonic.org domain as well and it works normally for me, though I had to manually set a redirection rule in my router to make it work with HTTPS. Keep in mind Subsonic will try to use uPnP to open up the port you have it running under, but there's a chance that may fail. If that happens, you need to set the rule manually on your router/firewall. Testing it with a browser should show if it works in general or not.
Finally, regarding your last question: you would lose any functionality not currently supported by the plugin. There are things I have in mind to add in the future, but it will always be limited to: a) what the Subsonic API allows and b) what MusicBee allows in Plugins.