Yes, both heavy and hard as in "difficult." But it doesn't have the physical meaning of hard, as in, "Cheese is soft but rock is hard."
Sure, and nehéz munka is a good phrase. I think it primarily suggests hard physical labor, although with context it could mean other kinds of difficult work too.
For German speakers: There is the same "problem" in German. "Schwer" means both heavy and difficult/hard (I admit there is also "schwierig" which in fact just cannot be used for heavy/weighty). But in German as well as in Hungarian, "schwer" / "nehéz" are often interchangeable.