During exams people sleep there all the time. My friend did lol. So to answer your question, yes you can most definitely sleep there.
As for showers you don't need a membership to use the showers, but I'm not sure if the building is open late at night. I'm not too sure about the lockers either but you can try and get one (I think registering for a locker is done in the campus bookstore), or maybe share with a friend? That's what I did last year.