The short answer is no. Activities and Resources can only be added to courses. I do not understand your third statement...when you create an assignment, you must give it a name. You do not have to specify the course name because it is already connected to a course. Unless you're trying to set it up directly in the database which a) asks for the course id and b) is a really bad idea. That said, if you want to set up an assignment on the front page, you can. Because Front page (aka Site home) is a course. Course id=1.
Go to site home. Go to Administration-> Front page settings->Turn editing on. Now you can add stuff to the front page.
Downside: unless you specifically enroll people in the front page, their role, by default is authenticated user, which may or may not allow them to use the stuff you build. Depends on settings and what you want.