> use your address firstname.lastname@example.org to send emails from mylittlemoodle.com
You mean Site administration / ► Plugins / ► Message outputs / ► Email: No-reply address (noreplyaddress) and Always send email from the no-reply address? (emailonlyfromnoreplyaddress) set? Yes, that will greatly simplify the mail sending because you have control over the DNS of your domain.
But some do not like that setting because the recipient can reply directly to the sender, not realizing that this is an impossible task, unless you've limited the mail addresses of users to your own domain(s).