How can I get the timestamp of the last time an email was sent by an action?
You can check the email sent time-stamps on Service Center Logs (Monitoring -> Emails). There you can see the emails sent by eSpace but not from a specific action. If you need to have that information, you can add specific audits on your applications with the Audit built-in function.
For accessing that kind of information on your own application you can reference the email tables by going to add remove references and then select the email tables you want.