So ... if I just close the browser, should I be logged in when I come back?

Yes. I'm the only one who uses my computer so I never log out of DW, LJ, JF, FB, Twitter... and not having to log out/in is so convenient, and awesome! (I'm almost positive I have "Remember Me" checked on all sites that have that option.) However, I'm not sure whether you'll remain logged in, sans "Remember Me" being checked, if you don't delete your initial log-in cookies too.