What's the process for using this for an entire domain?

I’d like to use Mail-in-a-box for an entire domain (domain.tld) that has mail.domain.tld for the Mail-in-a-box server and then other services, such as the main site are served on server.domain.tld. I used CloudFlare to manage DNS.

I’ve found that I get various errors on the System Status Checks that give me the impression it doesn’t really like this setup.

Are there any instructions that indicate how to set such a scenario up properly?

Some of those errors can be overlooked. Especially if you are using External DNS.

What services, specifically are you wanting to run? MiaB is designed to be a stand alone email/contacts/calendar project. As far as hosting a website, a static site is fine, but beyond that is unsupported.