It’s funny because the way I describe the concept of federation to people is to compare it to email.
“You know how you can email somebody and it doesn’t matter if they’re using Gmail, Yahoo, Exchange, etc? Just apply that concept to other services and that’s what federated means.”
So yeah you can host your own email just like you can host your own mastodon instance, Lemmy, Calckey, kbin, etc.
I’ve been a professional sysadmin for years, including managing email systems covering tens of thousands of users.
Don’t do it.
Find a host you like and just use that. If you have to pay that’s probably better, that (hopefully) means you’ll have a better privacy policy, no ads, etc.
Email as a concept goes back to 1981 with tons of bolt-ons and changes along the way. You’ll go years with everything working fine, then suddenly you’ll find yourself sinking a lot of time into it when something suddenly doesn’t work.
Plus you can have the most modern setup in the world then find yourself dealing with a client that’s using the default sendmail config and refuses to change anything, now you’ve got to create exceptions, blah blah blah.
I think most other spam filters can get pretty good at you mark mail as spam and train it.
No idea about push. IMAP does have the ability to keep a connection open and get notifications and some mobile clients support that. My guess is, if the provider offers an app they probably support pushing to it somehow.