for example, can I use the same domain name (myawesomecompany.com) for a custom domain name in Azure in order to set up email services in Azure for that domain?
Yes. It is possible to use email service in Microsoft 365 for your domain when having your company website hosted by other service providers.
To setup your domain in Microsoft 365 for email service, please follow the link below to verify your domain:
Add DNS records to connect your domain
Additional link about Domain FAQs which may be helpful: Domains FAQ
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