[whpalmer4]

No, it doesn't do completion dates. You could put the completion date in the note so you don't have to look it up elsewhere, but you'll still have to complete it by hand (if you bring up the inspector or add the completed date column to the outline, you can mark it complete later but still have the right time/date, if that is important to you).

If you are interested in doing some Applescript coding, I believe it would be straightforward to modify the stock Omni script that does the mail handling to immediately mark the newly created action complete as of the time it was received and processed. You could set up an additional filter rule in Mail to divert the messages intended to add completed items (you could send them to a different address, for example, or perhaps start the subject line with "Completed:") to the modified script.

A no-tinker approach would be to forward the email upon receipt, do the task, mark complete :-)