Hello, the Red button just closes a window, it doesn't quit Mail in anyway, it sounds like you may have a Mail stuck trying to go out???
Hold CMD+alt+esc keys to bring up the Force Quit menu, choose Mail.
Let's see if this helps to start with...
Safe Boot, (holding Shift key down at bootup), run Disk Utility in Applications>Utilities, then highlight your drive, click on Repair Permissions, reboot when it completes.
(Safe boot may stay on the gray radian for a long time, let it go, it's trying to repair the Hard Drive.)
"Open MAIL, open Preferences, go to accounts, highlight your apple email account, near the bottom of this page is a box TLS. The default is blank.... this is wrong and should not be the default. open the drop-down menue and pick com.apple etc etc. Now save"