With EteSync you can share calendars and tasks. Apps like Tasks.org can communicate with EteSync (on Android at least, i don't know how the support is on iOS). The devs also provide a (Web UI, which can also be self-hosted.
The main feature of EteSync is end-to-end encryption, you can also self-host it.
Tasks.org is great. I use it with a CalDAV server.
I suggest Vikunja. There isn't a mobile app to my knowledge, but the web UI on mobile and desktop is very responsive. It should cover the rest of the items on your list
There's an app. It's still pretty early though, so they haven't put it on any of the stores yet.
Excellent, I'll check this out thank you!
There's an app, just not published to any of the stores yet.
Microsoft ToDo or Zenkit Tasks. Neither are self hosted but they work perfectly with multiuser setups.
Vikunja is the only viable thing. CalDAV spec allows to assign tasks, but we got XMPP moment here - spec exists, but almost no apps implement it properly.
Also you may take a look at https://github.com/awesome-selfhosted/awesome-selfhosted#task-management--to-do-lists
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