Use Cloudflare or PorkBun.com for cheap, no bullshit domains. As for the email host, self hosting not recommended. It’s a long battle to be not blocked by every other provider.
I recommend purelymail.com - no cost to add (even multiple!) custom domains, unlimited users, only pay for mail usage and storage. Go for advanced pricing until it starts costing you more than $10/yr. (Which it shouldn’t if it’s just you. Seriously this thing is cheap!) I just passed my one year anniversary with PurelyMail, and have spent $6 so far. This is my most expensive month, 85¢. And that’s only because I host a public Lemmy instance (small) and we had a few hundred spam signups which sends an email each time.
This will give you a total yearly price WAY under what Google or Microsoft will give you. Google is like, $7.20/user/month.
And if for some reason that service goes down one day, as long as you still have a mail client with your email stored in it you should be able to just switch providers and import your emails from your client. Make some backups.