I thought we might want to move the theoretical discussion out of the Worlds thread into one of its own where it's easier to find.
I don't have time to offer any answers right now, but here are some questions to ponder:
Thanks for starting the thread, gkelly. Some of the answers necessarily involve a bit of repetition of comments on the previous thread, but I'll try to minimize that. For those who prefer short and sweet, I'll start each response mono-syllabically

:
Are beauty and artistry synonymous when it comes to skating, or can you have one without the other?
Yes. If one agrees, at least on some level, as Keats does ("Beauty is truth, truth, beauty..."), with the classical concept of the underlying unity of the virtues, the identification of beauty and artistry becomes at least intelligible to a modern reader. My own personally held critique of the ancients (which is in fact the Modernist critique) is that, while the idea is philosophically sound, they refused to follow it to its ultimate conclusions in many of their arts, particularly in the visual arts.
The Greeks knew very well (as shown in their plays) that truth is often the opposite of "cutesy-wootzy" (as Mathman would put it). A more modern, very profound example that is very much in their tragic spirit, IMO, is this comment by the Confederate general Robert E. Lee: "It is well that war is so terrible-otherwise we would grow too fond of it." It is an old, paradoxical human reality that some of the things we find to be most truthful, and hence, beautiful, are often displayed in their most distilled form when we try to kill one another (courage, sacrifice, idealism, even love), but which can only be obtained at almost unbearably high cost.
This explains the appeal, it seems to me, of artists such as Kandinsky and Francis Bacon, who are, particularly when first seen, visually shocking in their presentation of the darker side of emotional and human truth. Unlike the Greeks, their technique is to demand that visual euphemisms that shield the viewer from what we find disturbing be cast aside. The unspoken manifesto and challenge: if beauty is truth, then be prepared to throw away emotional complacency. Now, I personally wouldn't hang them in my bedroom (they don't exactly help in getting to sleep). But it's a bit like how I feel when watching tarantulas on Animal Planet; it looks creepy and repulsive at first, but I find myself fascinated, and then begin to recognize how beautifully its features are adapted to its own needs.
So, yes, by analogy, the artistic in skating is always beautiful, but in principle can be unsettling, and doesn't always have to be "pretty", in accordance with the needs of its themes.
Can skating with no music be art? Does the presence of music demand evaluation as art even if the skater does a bad job of reflecting the music or uses it only for counting?
Theoretically,
Yes (but with overwhelming practical qualifications, so actually my answer would be
No. OK, I cheated with the monosyllabic answer

). Quite a while ago, I gave the example of a hypothetical program set to (if that's the right way to put it) John Cage's four minutes, and change, of silence (officially named 4'33", but rumor has it that Cage said so to his agent in sign language, so who really knows?

Of course, it may not be completely silent if you count crickets chirping and the shuffle of heels creeping for the door). It can be done. It might even be done with intent. It is likely to be done as a daring experiment, in the avant garde spirit. It also likely, IMHO, to be an interesting, if daring, failure (the way that most mutations in the scheme of evolution are failures).
I view musical accompaniment as one of the most important tools in helping the skaters do their aesthetic jobs. The practical advantages in executing many aspects (particularly choreography and interpretation, as well as overall expressiveness) are so obvious that I won't elaborate further. The precision that can be achieved with the aid of music is not easy, perhaps impossible, to replace.
Layfan gave the example of Baryshnikov performing to his own heartbeat. While I imagine that must have been the definition of bravura, I also think that an intellectual curiosity to push the envelope, and perhaps even the prideful desire of a supreme performer to demonstrate that bravura, may have been the primary motivations. Was it a fascinating experiment and worth trying? Probably it was. Would Baryshnikov, and audiences, have said that this was among the richest and most emotionally meaningful of his performances, that they'd regularly pick if given only one choice? I would guess not (though I'd be interested in Layfan's views, having actually seen it).
Unless you have a very compelling reason and purpose, the last tool you'd want to leave out of your kit is the one whose uses are so ubiquitous and malleable, the monkey wrench of figure skating artistry. You can try to build things without it, but why would you want to?
Is show skating all about art and not a sport at all? Do cheesy shows count as art or only hoity-toity ones?
Yes, at least in its ideal, if the answer has to be one or the other. It isn't sport unless there is the competitive pursuit of an explicit, physically measurable physical goal (this definition is deliberately designed to exclude not only ballet performances but also things like chess, contract bridge, and e-games like Halo).
The caveat is that the artistry of a show very much depends on the skater's attitude. Some will see it as a way of simply performing at a more relaxed and less stringent level after a long season, while picking up some cash in the process. It might still be art if one really stretches the point, but of, shall we say, dubious quality.
The opposite attitude can produce startlingly different results. My favorite Mao performance to date is her Por una Cabeza ex. There was a delicate naivete that was at the same time alluring, a freedom of expression with an insouciance of technique, that was captivating, not least because it was emotionally true to herself. By way of contrast, the Bells program, while a brave attempt to stretch her boundaries, was ultimately unsuccessful artistically IMO, not through lack of ambition, but because it lacked that ring of emotional self-awareness and truth. I'm not sure exactly why Mao's ex programs of late have often been artistically more compelling than her competitive ones, but there it is.
Yuna's shows, on the other hand, have often been aesthetically a mixed bag for me (and I say this as a very great fan of hers), although her competitive programs have been artistically of the highest quality in recent years. In my view, her Meditation de Thais performance at the ATS Summer show (for which there is a gorgeous vid on Yuna's official youtube channel) was artistically the most hypnotically commanding, nuanced and assured performance of any ladies program of the past cycle, competitive or ex. On the other hand, I was very disappointed with Bulletproof; despite the anticipatory buzz, I thought it was an emotionally lazy program, perhaps a result of a too conscious attempt to appeal to a wider fan base. I have high hopes for her new competitive programs, though.
Your last three comments are out of my bailiwick.