How Does Outlook Profile Information Work for Custom Domain Email Accounts?

• Where Outlook pulls profile details from for a custom domain email
  • If the account is set up with Microsoft Exchange, once you add the account, it will be default download 1 year of email and all the calendar and contacts. 
• Whether profile information is managed in Outlook, Microsoft account settings, or Microsoft 365 admin settings
  • Outlook can have many different profiles, you can change which one in the mail app in the control panel to set as default. The profile information is in each profile, the accounts added and the OST file location.
• Which profile details are visible to recipients when sending emails
  • No details should show but your email account you're sending form
• Whether there is a way to personalize the account profile for a more professional appearance
  • Depends, what would you like changed? Everyone using Outlook Classic has the same look/feel. You can change the colors for yourself, not for others to see unless you change the font colors. 

Which profile information are you referring to and in which context?
You mean the Contact Card?

When you are sending an email, you are not sending any profile information to the recipient aside from your display name and email address. 

If no one picks up this question here, you might be better served by posting this question in the Outlook Q&A Forum:

https://learn.microsoft.com/en-us/answers/tags/778/office-outlook

This presents your question to a much larger audience and gets you access to the Outlook Subject Matter Support Specialist who monitor that forum.