For FastMail I had to use SSL on port 565.
I normally use TLS or STARTTLS on port 587 from email clients, and I believe port 465 should work too, but no combination of these would work from PrestaShop.
It didn't make any difference for me, if the Customer Service email address (Shop Param > Contact) is the same as the site domain or different.
It didn't make any difference if the 'Mail domain name' (Adv Param > Email) is the same as the site domain or different.
Hope this helps someone.