Hello.
-Which version of OS X/macOS?
-Which browser/version - Safari, Google Chrome, Firefox?
The browser issue could be a couple of things:
- > System Preferences... > General > Close windows when quitting app isn't checked
-The pages that are loading are set as startup pages - especially in Google Chrome and Firefox. Chrome > Preferences > On startup > Continue where you left off or Open a specific page or set of pages
Mail issue:
-Do you mean an actual Notification Center notification in the upper right of the screen?
-Is it a Gmail account that is doing it?
-Could you possibly have email forwarding enabled on one account, that is forwarding the emails to another?
C