Would it be possible to have my library online (dropbox/ private server) and use it just like Tidal?
That would a great feature but its not available. I’ve been using Subsonic for 5 years now to access my music from the internet. It works very well.You can even create profiles and let others listen to your library. There is a phone app that lets you stream or save music locally. Overall, I highly recommended Subsonic.
Are their any subsonic iOS apps that support headphone crossfeed?
I dont think so but since Subsonic allows you to cache music locally to your mobile device you can use another app to listen to music that has the crossfeed feature you are looking (I am not a headphone user so I do not know for sure if there is an app that does this). I personally use Poweramp to listen to music I cache using Subsonic. Note that there are mutliple mobile apps that can be used with the Subsonic server running on your pc at home.
Unfortunately: iOS apps are sandboxed. Anything downloaded to an app isn’t available to other apps. Worst yet Apple doesn’t allow 3rd party apps add music to the default iOS music library, you have use Apple Music or crapTunes.
Looks like Vox Music is still the only solution for iOS that supports sync to device and has crossfeed built in. Unfortunately they force you to buy their cloud subscription to have that sync capability.