Hmm; I use Thunderbird as my email client and run FIVE email addresses from it.
The major email account is on Roadrunner.com and another is on Gmail.com
I can send/receive on any account -- even on my smartphone.
JB: His problem is that he owns his own domain and he wants to get email from it and anyone who would host mail for him would charge him. He is looking for someone to do that for free.
The part you are talking about is referred to as Client software. So that's Thunderbird, Outlook etc..those are the programs you use to connect to your domain with your credentials (login and password) to retrieve your email. In your example you are getting email from the Roadrunner domain (Your Internet Service Provider) and Gmail (Google).
Scott wants email from his domain @aaahawk.com. For that to work he has to hire someone to host his mail like:
Plans start at $3 a month. 30-day Free Trial. All the Control and Features you love with no creepy surveillance. Stop paying for email with your privacy.
Access business email, calendar, and contacts on your PC, phone, and web browser with Exchange Online. Hosted email that’s easy to use and easy to manage.
www.microsoft.com
Then he has to go in there and manipulate the MX (Mail Exchange) records to tell the world's DNS (Domain Name System) where he is when someone sends mail to someone@aaahawk.com
I think you can find someone to do that for a low cost but I don't think for free.
By the way I use Google for this solution and it most certainly not free but I consider the Google/Gmail system the best at email over all devices.