You can’t use the iOS Mail app, there’s no POP/IMAP support. You have to use their proprietary app.
Notifications.
Who uses notifications anyway?
Are you serious though ?
Yes. I do not want apps to be proactive. If I need to check mail, I will do it. If I need to check it regularly, I'll set up a calendar event (OK, once in a while I need notifications), but these should be notifications I personally configure.
Mail app.
Maybe true that a lot of people turn off banners and sounds, but even a badge to show you have x unread messages needs the notifications. Else u have to open the app to check ..