When philosophers talk about qualia they are typically discussing the mind-body problem and the prospects for materialism. They are not interested in the general theory of qualia. They may cite some examples of qualia, but they don’t enquire into the general structure and function of qualia—as a linguist might study the structure and function of language. There is no “qualistics” analogous to linguistics. And indeed the subject is difficult and obscure, even more so than the nature of language. Here I will merely tickle the subject of qualia in the hope of eliciting a gurgle that might reveal the general nature of the beast. The analogy with linguistics provides a useful starting-point, because quite a bit about language has been discovered, particularly in the post-Chomsky period. The most basic fact about language is that it consists of a finite lexicon that is capable of infinite expansion by means of the rules of grammar. Language has an elements-and-rules structure with infinite potential. Grammars generate an infinite array of expressions by operating on a fixed and finite set of linguistic units. Is something similar true of qualia?
The plural form of the noun “qualia” suggests a stock of distinguishable discrete elements, and indeed introspection confirms that our sensory consciousness is constituted by a totality of just such elements—sensations of color, feelings of pain, tastes, smells, etc. These come in enormous profusion: it would be hard to put an upper limit on the number of qualia accessible to a typical human perceiver (or other animal). Just consider the number of distinct qualia you experience with a closed eyelid! I would suspect that qualia vastly outnumber words. So we may state the first law of qualia science: there exist a colossal number of qualia in the universe. The number is presumably finite, and is probably less than the number of stars, but it is truly massive. Moreover, qualia, unlike words, vary continuously, being more analogue than digital (just consider all the shades of blue): so the infinity of the continuum applies to them. The cardinality of the qualia lexicon is astronomically large. In addition to this the “grammar” of qualia is immensely productive: qualia can be combined ad libitum, both in space and time. Complex totalities can be formed that are completely novel and will never be repeated—as when you visually take in a new scene of any complexity. At any given moment the five senses are saturated with seething populations of qualia, which together yield an enormous array of variegated qualia ensembles. The brain in all its complexity must be responsible for this vast qualia universe, by virtue of mechanisms little understood. Consciousness is constitutionally capable of housing these gargantuan combinations of individual qualia. The brain is clearly an incredibly complex and intricate qualia-generating machine.
The second law of qualia science states that qualia are extremely various. Each sense has its own proprietary range of qualia, and within each sense there is also considerable variety. There are a great many qualia universals. Qualia types are evidently more various than physical particle types (which are relatively few in number). Words differ in their types too, but qualia fall into far more categories. It would take a determined theorist to be a qualia monist (or an especially blind one). The qualia of different senses don’t mix, despite their freedom of combination within a given sense: you can’t see red and taste pineapple within the same sensory field. Qualia shun each other when they are too different, though they congregate happily when they share a basic phenomenology. Perhaps there could be olfactory eyes that respond to light and chemical impingement at the same time, thus producing a combined olfactory-visual field; but that is not the set-up on planet earth (so far as we know). So there is a definite limit to the combinability of qualia into unitary percepts, even though within a sense modality there is great plasticity (not to say promiscuity). Qualia are like members of a tribe that will hang out with any member of their own tribe but will not mix with members of other tribes.
But there are some strange borderline cases and apparent exceptions, which bring us to the heart of qualia darkness. For one thing, qualia enthusiasts never cite impressions of geometrical forms as instances of qualia: it’s always sensing colors not sensing shapes. Do we not have shape qualia? But surely we do perceive shapes, so there ought to be corresponding qualia. The reason for this omission, I suspect, is that we experience shape with more than one sense—with both touch and sight. So there is nothing qualitatively distinctive about sensing shape. Yet it does seem as if a single experiential type is involved in sensations of circularity (say). This is why it is tempting to give an affirmative answer to Molyneux’s question: yes, a blind man made to see would recognize circles based on his previous tactile experience with them. So here the quale seems to hover between sensory specificity and abstract generality. We don’t really know what to say about geometrical qualia—that is, qualia corresponding to the traditional category of primary qualities. This is why all the examples cited involve secondary qualities. But the question cannot be avoided: do we, or do we not, have qualia of what are called “common sensibles”? If we do, qualia can hop between the senses; but if we don’t, we are qualia blind to certain qualities of things.  The question remains infuriatingly obscure.
Obscurity mounts when we consider synesthesia. It is said that red resembles the sound of a trumpet, and some people experience letters of the alphabet as having certain hues. Is there, then, a type of experience that straddles the two—a type of quale that unites these disparate qualities? Are redness and trumpet sounds instances of a common qualia type for people who sense the resemblance? Are there such higher-order qualia? The idea seems not without merit. This suggests a hypothesis: that there are qualia that transcend specific concrete qualia but coexist with them. These unifying qualia hover in the background somehow, permeating our more immediately discernible qualia.  Maybe we have no words for them, but they are real nonetheless. Concrete instances may be variations on a phenomenological theme. Could it be that there is a single basic type of quale that gets specialized in the different senses, producing variety from uniformity? Can we imagine a sentient being whose entire qualia space arises from a subset of uniform qualia originating in a particular sense? Suppose that in this being vision evolves first and then the other senses piggyback on it by modifying the qualia associated with vision, thus producing qualia called by other names. The sensation of red is coopted by hearing to produce the sound of a trumpet, say.  Thus there are qualia universals that span the different senses—rather as there are linguistic universals that span different human languages. The variety we observe arises by a process of transformation amounting to metamorphosis. This seems like a strange idea, to be sure, given the qualitative differences between qualia across the senses, but can it be ruled out a priori? Might the variety of qualia be less than appears to casual inspection? Certainly we have repeatedly discovered deeper uniformities in nature than strike us at first sight. Concepts are not sense-specific—they don’t have a sense-modality written into them—so perhaps qualia are also more modality-neutral than we suppose. Could there be unconscious qualia that underlie the conscious kind available to introspection? Could the apparent heterogeneity disguise a more basic homogeneity? Might qualia admit of a distinction between surface structure and deep structure? We don’t know, but the question seems real enough. We are certainly familiar with the idea that the introspectively available part of the mind is only one part of it. Qualia might enjoy a substantial unconscious life in some form or other.
It may be said that the analogy with language is imperfect because the principles of qualia combination are nothing like grammar in the literal sense. I would agree with that—indeed, insist upon it—but this is a dispensable feature of the analogy. A better analogy would be bird song, which has the advantage of locating qualia at a lower evolutionary level: bird song is also built around a set of primitive elements and rules of combination, species-specific and quite intricate. Also, the notes of bird song vary continuously like qualia, thus producing enormous variety. Architecturally, then, individual qualia resemble the elements of bird song; we can readily imagine a mapping from one to the other. The basic brain mechanisms that allow one might allow the other (not that we know what these mechanisms are). From a computational point of view, both are functions from primitive elements to combinations of elements, possibly exhibiting hierarchical structure. Qualia do seem to cluster and embed, forming gestalts, dividing up the world into useful units for effective behavior (just consider your visual field right now). We have no idea how qualia evolved, singly or as combinable elements, but presumably mutations produced machinery capable of generating their distinctive form—a kind of program for qualia manipulation. And presumably this elaborate mental capacity has survival value: qualia help keep the organism around. The genes wouldn’t go to all that trouble for no reason. Qualia are psychologically real, cerebrally embodied, and functionally adaptive—they have to be. They must be as natural as bird song, and analogous to it (up to a point).
It should be noted that qualia are not confined to the senses. They also appear in the imagination: our mental imagery is qualia-laden. Here they combine in new and surprising ways, following the elastic rules of image formation. They take on fresh combinatorial possibilities, not being stimulus-bound. Thus we have the qualia-borne imagery found in dreams: boundless, bizarre, creative, and anarchic. Dreams are qualia heaven—the place where qualia go to do whatever they feel like doing. And they are so nimble, so mischievous, and so free. But even in daydreams qualia flex their creative power, fluidly combining and recombining, always resourceful, preternaturally agile. They form the inner speech of the senses. They also feed into the thought process, though they seem shy about actually entering that domain (we don’t “think in qualia”): what we think and believe is shaped by the qualia populating our sensory consciousness—colors, sounds, smells, tastes, feels. So qualia fit into the overall economy of the mind in characteristic ways; they aren’t just marginal. Again, they are like words in this respect—here, there, and everywhere.
I have tried to avoid the vexed question of the ontology of qualia. What are they exactly? Are they representations of external properties or are they qualities of consciousness itself? Are they non-physical? I will simply say that they are subjective-objective hybrids: episodes of seeming to apprehend qualities of things. The quale associated with seeing something red is a seeming to have a red thing before you: it is thus a synthesis of a subjective state and an objective quality (in one sense of “objective”). Not that any of this is easy to understand, but it is the right way to describe what is at issue. Accordingly, qualia are instances of such episodes of seeming—that is, states of consciousness in which a quality is apprehended. These are the things that exist in such profusion and that get combined so freely. If it is any consolation, it is also not at all clear what a word (lexical item) is: is it a sound wave, an act of speech, a mental entity, an abstract pattern, or a brain configuration? The ontology of words is also obscure (ontology often is: numbers, meanings, beliefs, even physical objects). But this obscurity shouldn’t prevent us from recognizing important truths about the subject matter in question, particularly its structural and functional features. As with so much about the mind, we are only at the beginning of understanding it, and may never achieve the understanding we seek. 
 People sometimes say we perceive visual shape and tactile shape, but never just shape: but this is an obvious cop-out—surely there is something in common between the two, something with phenomenological reality.
 Sartre would say that every quale carries with it a sub-quale of nothingness, since nothingness is the essence of consciousness: in seeing something red, say, the subject apprehends the emptiness of consciousness itself. A referential theory of qualia would likewise imply that every quale embeds an awareness of an act of intentionality common to all qualia. In both cases we have cross-modal qualia of a rather abstract type.
 We can formulate a synesthetic version of Molyneux’s problem: would a person born deaf recognize the sound of a trumpet as similar to red if suddenly given the ability to hear? That is, is this similarity inherent in the red quale and capable of extrapolation? I myself sense a strong similarity between whiteness and silence, and also between hot peppers and hot objects (the use of “hot” for both is surely not an accident). Thus there might be a universal phenomenology (across the senses) analogous to universal grammar (across languages).
 It is an interesting historical fact that the word “qualia” and the associated concept have been around for over a hundred years (C.S. Peirce introduced “quale” in 1866 and C.I. Lewis introduced “qualia” in 1929), but the notion is still highly controversial even as a descriptive term. The subject is shrouded in obscurity, and hardly exists as part of scientific psychology. It awaits its Noam Chomsky to set it on a scientific footing. We barely have even a superficial taxonomy of qualia. Qualia theory is like the linguistics of the Stone Age.
Maybe we can’t get at what is in common between shapes, as per your first footnote. I’m thinking along the lines of Wittgenstein’s point about logical form; propositions representing reality, but not necessarily being able to represent what they have in common with reality in order to represent it.