The Mail app runs constant fetching in the background? I don't think so.
That's not how push works, no (fetching would be pull, not push).
The Mail app has a constantly-running background process that keeps a connection open. The server can then push data as it comes in.
As I wrote. And some 3rd party email providers (Fastmail, Mailbox.org) can send APNS notifications to the Mail app.
They can, but that doesn't help third-party apps.
Of course they can. Both Spark and Airmail on iOS support Exchange ActiveSync, and the Fastmail, Protonmail and Tuta apps supprot push notifications. The issue is that it requires support by the email provider, but that is true for the stock Mail app as well.
How do Spark and Airmail push stuff to your iPhone? By having a server of theirs knowing your credentials.