Hi, @Craig Richards
About adding business email to M365 and working in Microsoft Exchange, you can refer to the following article.
Add a domain to Microsoft 365 - Microsoft 365 admin | Microsoft Learn
Connect your domain to Office 365 - Microsoft Support
Regarding the search function in Outlook, do you mean that the target user's email address cannot be found?
In my environment, searching for a target user's first name will automatically search for their email address.
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