On Android, the PWA should work very well. Multiple staff members use PWAs daily.
On iOS, sites hosted by Discourse get full push notifications in DiscourseHub. Self-hosters get push notifications via polling, yes.
Unfortunately, the DiscourseHub polling for self-hosters relies on system background tasks. They are not guaranteed to run at any particular time… so, nobody knows what the delays are. The OS decides. If you use the app regularly, I think (with very little certainly) that the polling happens roughly every 15 minutes. If you don’t use the app regularly, it can be considerably less often.
No, it should be for topics, chats, PMs.
A webview wrapper app has a high chance of getting rejected, yes. A full native app, probably not. But a full native app is considerably more complicated to build.