Quoth Wikipedia:
The Greek word khthon is one of several for "earth"; it typically refers to that which is under the earth, rather than the living surface of the land (as Gaia or Ge does).
Now, I know the term "Gaia" not only denotes the "living surface of the land", but also a personification thereof, sometimes referred to as a "goddess", but I'm not quite sure whether that is a suitable term at all, since, while certainly a mythic figure of divine proportions, Gaia was not subject of worship, though I may stand to be corrected regarding that matter. In modern European dictum, there is no real differentiation between surface and interior when speaking about Gaia as the goddess of "the Earth" - this always seems to imply that, though the ancient Greeks may not have held Earth to be structured as we know it to be, they would have believed all of planet Earth, everything that was soil, rock, and so forth, to fall under the domain of that personification, Gaia. However, given the passage quoted above, did the ancient Greeks see the personification, qua deity, Gaia only as a personification of that surface, excluding the subterranean khthon? Or would that also have fallen under what Gaia represented? If not, was there a personification or deity of khthon itself? Was khthon Hades' domain, or was Hades' domain perhaps located within the physic khthon, or was it considered to be a place below khthon, or even entirely separated from both the surface and the subterrane?
This is a great set of questions, and unfortunately there aren't very satisfactory answers to a lot of them. Part of that is from lack of good evidence, but I suspect also that some of the divisions you are asking about were not entirely clear even to an ancient Greek.
First, Gaia (or Ge) herself was often personified, and did in fact receive cult (unlike many other "chthonic" powers). She is the same as Khthon, technically, and the two are never obviously separated as far as godhead. In Greek, the word χθών (khthon) is generally reserved for the area beneath the surface, below the "Earth." It is most often used in association with chthonic powers, which were separated from "upper" powers in matters of cult practice (though this is now thought to be an old-fashioned way of thinking about Greek religion). A translation of Burkert's Greek Religion would be the place to look for more on things chthonic. Gaia, when she is perceived at all, is generally everything, but perhaps most often the "upper" or "surface" Earth.
There were cult statues of Gaia in antiquity, though unfortunately none have yet been found. She was often given cult in conjunction with other powers (like in a temple to Demeter at Patrai), but sometimes alone. She was honored at Athens, Delphi, Sparta, Olympia, and many other places, usually in her guise as the mother of everything.
That seems to be the limit of her identity, however. Hesiod, in his Theogony, tells us Gaia is the offspring of Chaos, along with Tartaros, and the mother of her consort and equal, Ouranos. She is everything that is not the Pit of Tartaros or the Sky, which blankets everything. That means that technically, yes, she encompasses the Underworld within herself, though I cannot call to mind a single instance in which that notion is explored. Despite the cults to her in widespread places, I would not suggest she is a goddess who is particularly active in day-to-day, the life and death, of the mortal world. If the Greeks imagined that she was aware of the living upon her, or the dead within her, we hear nothing of it. Hades was lord of what was below--the darkness and the shades of that darkness, but his domain was still clearly a part of the boundaries of Earth and Sky. It was through Hades that you approached the Pit of Tartaros, and indeed heroic travelers there could visit those terrible gates. So when Aeneas is down there:
Aeneas suddenly looked back, and, below the left hand cliff, he saw wide battlements, surrounded by a triple wall, and encircled by a swift river of red-hot flames, the Tartarean Phlegethon, churning with echoing rocks. A gate fronts it, vast, with pillars of solid steel, that no human force, not the heavenly gods themselves, can overturn by war: an iron tower rises into the air, and seated before it, Tisiphone, clothed in a blood-wet dress, keeps guard of the doorway, sleeplessly, night and day. Groans came from there, and the cruel sound of the lash, then the clank of iron, and dragging chains. Aeneas halted, and stood rooted, terrified by the noise. (Vergil Aeneid VI).