How to stop the Mail app from running every 10 minutes on the Mac?
This is not normal behavior. Edit your question with your console log information from the moment of launch, it might help.
Notification center will provide alerts when mail arrives if set to do so, and clicking on a message will launch Mail. Mail messages also reside in the notification center and clicking one there will also launch mail.
Short of these events, Mail should not launch itself, unless you have some background process or script you are unaware of.

There is a background process or helper to allow the notification center to show incoming messages while Mail.app itself isn't running. This is normal behavior. If Mail is turned off in the notification system preference you won't have any intrusions.

Of course different things such as shell scripts, other apps and so on can launch apps so if you've configured other apps to launch Mail you need to deal with that scenario. This is where the log files can give you a hint as to what happened immediately prior to Mail launching.