Can I save my own Email Message Templates

The Message Templates that you see in Settings >> System >> Message Templates are the ones that have an automated system trigger to go out - as such, you can't add to that list. 

However, if there are certain messages you send again and again, you may want to save those in a Whiteboard for easy copy-paste access.

If you have specific emails that go out based on a cadence for clients you can put them in a Course to go out based on the schedule you decide.