1 -- I don't know. That's just how the software is made. If you choose forever, you'll be logged in for a very long time. It's not really "forever", or "always", because every now and then, when we work on the site or the server, everyone gets logged out. Or sometimes you might want or need to log yourself out.
But if you'd rather only be logged in for each session, choose a very short time when you log in. Whatever the amount of time you choose is the life of the cookie. I'm not positive, but I think if the cookie expires while you're still logged and using the site, you stay logged in until you leave the site. I don't think it logs you out the moment the cookie expires, if you're still on the site. But I could be wrong....
I think if you click the Login tab, in the navigation menu, instead of using the quick login in the top-right corner, you can enter zero in the "Minutes to Stay Logged In" field. I haven't tried it, but it looks like it might be possible.
2 -- I'm not sure if I completely understand what you're asking. The cookie is a small bit of code that the website gives to the user (user's computer). And the website has communication with the cookie about how long you want to stay logged on. But I don't think there's anything specific to you, the user, that's kept on the server. Well, not related to the cookie anyway. Of course your user account (email address and password) remain until the website closes or you delete your account. Once the user deletes their cookie, the website can no longer communicate with the user's computer (about log in time, in the case of session cookies).
Besides the options when you log in, about how long you want to be logged in, there are usually a Lot of options in your browser about handling cookies. I guess most modern browsers can delete all your cookies when you close the browser. Once the cookies are deleted, the website can no longer communicate with them. The next time you visit or log in to the site, it will offer you new cookies. And you can set your browser whether you want to accept them or not.