Pop is download only (and optionally delete on server when downloaded), IMAP is sync so read states, moves, deletes, etc., will sync to the server and between all clients.

But both exist still, just most email providers want you to leave your email on their servers so they can charge you for storage, scan it for advertising, etc., and most people want the state to sync across devices, so a lot of places just don't implement it anymore and just allow either web-only or IMAP-only (usually paid since they don't get to show you ads).

And yes you can set up a local software to download email from another server offering POP or IMAP and then either use a client application or self hosted web application to view them. This is not too common so it might be harder to find instructions, but it's doable.

Which part are you concerned with. The biggest issue with email servers is trust. Many companies will refuse to connect to your SMTP server to send you emails or reject your emails your SMTP server is sending to them if you aren't on a whitelist or if your IP address has been used for spamming in the past and it's on a black list. The easiest way to get around this is to subscribe to an SMTP proxy service.

The other issue you may find will happen no matter if you selfhost or not. When using a custom domain, some servers have whitelists of domains in addition to IP addresses and if you aren't on it they will silently reject emails you send or that they were supposed to send to you. I have a couple of places (health care and financial companies mostly) who do this, so I have to use gmail, which ironically means the email is less secure, especially with Google's recent changes in policy about reading your emails.