They can be, but that's not typically thought of as their main purpose. You also wouldn't likely be living on the inside surface with the star overhead, but rather within the shell or on its outer surface facing space. The inside facing the star would almost certainly be reserved for harnessing the radiation.
Yes.
That's the general gist of it, yes. If you're looking for habitable spaces for people, you'd be vastly better served with large numbers of O'Neill cylinders or similar smaller stations. The DS is meant to encase the star to harness 100% (or nearly, you'll never get it perfect) of its energy production. In some sci-fi I've seen it can also be used as a means to "hide" the civilization from the wider universe, though they tend to get handwavey with how exactly they manage to sustain such a perfectly closed system.