You are mixing the "session" as the current identifier of your browser/you in a web application, and the session of the user in her computer, meaning all the tabs opened.
Now for your other questions :
If I stop the computer with my web-app, and then start it again before
the timeout set on the session on the server expires, then anybody
that starts the computer and clicks "restore session" could continue
in my web-app where I left off.
It depends on how you defined the session duration & expiration. You can define the session to expire when the browser is closed. If done so, the user won't be able to restore the session when he'll reopen it's browser. Now if you defined a duration instead, it depends if the session will have been expired or not.
Is there any way to kill the sessions after I close the browser like in IE and Google Chrome?
Like I said, just define no expiration date for the cookie session (if you use it like this) to expire when the browser is closed and it will be good.(see Wikipedia for that.)