I would like the ability for emails regarding items in SharePoint having been created or modified to come from firstname.lastname@example.org or some generic email with the SharePoint site's name showing as the sender, just like in SharePoint Designer, instead of these general notification emails showing up as coming from the Flow author (i.e., me). This is for the "Office 365 Outlook Send an email" action, unless there is another action I can use for that.
We not plan on adding this feature, but you can already accomplish something almost identical to this via two different mechanisms. If you want a generic email like email@example.com, that's supported via the Mail - Send an email notification action.
This connector works the same as the normal connectors, but instead of using a user account it acts as its own Flow based connector. Therefore when you send emails using it, the message will read as coming from firstname.lastname@example.org and display like the following:
Alternatively, if you want it to come from an email inside the domain you can do that by using a Shared mailbox, as called out below in the comments.