Also check out funkwhale!

Funkwhale is awesome, but it's also one of those FOSS projects whose website doesn't exactly illustrate what it exactly is or what it's exactly for, so here's a direct link to straightforward answers for both of those questions

https://docs.funkwhale.audio/features.html#scope

I want to say +1 for funkwhale, but I didn't get it to work with the subsonic client and haven't found yet a way to download/cache songs from my library on the phone. Have you figured this out?

Subsonic phone clients were kind of a crapshoot last I checked. Ultrasonic was the only one that kinda worked for me.

https://github.com/ultrasonic/ultrasonic

Ahh no I haven't, if that's not working then it might be better to stick with subsonic...

Airsonic is also nice. I was a long time (paying) Subsonic user, but got a bit fed up with it and migrated.