I ran across this paper today (via languagehat): "The indeterminacy of word segmentation and the nature of morphology and syntax", by Martin Haspelmath. It's a bit technical, but the boiled-down summary is that it is difficult to define "word" such that the concept is applicable and useful to all languages. (Some linguists might disagree; I tend towards sympathy with Haspelmath's view.)
So there is no "official or commonly understood sense" that we can appeal to here -- or rather, there are lots, and which one we choose depends on what we want to do with it. One simple pragmatic rule of thumb would be to say that a "word" is what's in the dictionary. In English, this means that things like "New York" and "neoclassical" are indeed words, and in Japanese it would mean that the commonly accepted kanji compounds are words and the rest are not. But I guess you don't like this answer, because otherwise you wouldn't have needed to ask the question.
If I combine kanji in some new and useful way, have I invented a new word, or am I merely using the kanji in the Lego-like combinative way they're intended to be used?
The second half of this sentence kind of begs the question, I think: the way they're "intended to be used"? (Intended by who?) What I would say is this: when you combine kanji that aren't habitually combined that way, you are, in most cases, inventing a new word. The reason for this is simple: simply cramming Sino-Japanese morphemes together like that does not result in an unremarkable sentence, even if your meaning is crystal clear. People will notice what you're doing and consider it unusual. Thus, it's more akin to saying "blueplate" than "blue plate" in English. And this is leaving aside any consideration of accent, etc. (Note: I'm using "kanji" here but of course what I mean is "morpheme that is associated with a particular kanji".)
(Sidebar: We might want to exclude cases where you combine groups of kanji that are already words in their own right, e.g. 階級 "class" ＋ 意識 "consciousness" = 階級意識 "class consciousness". On the other hand, we might want to consider this a long-ass four-kanji word; just because we segment it into two words in English doesn't mean it has to be that way -- in German it's Klassenbewusstsein, which is clearly made up of Klasse(n) "class" + Bewusstsein "consciousness" in the same way.)
Now, some morphemes-that-happen-to-be-associated-with-kanji are "better at" making new words than others. That is, you can attach 新- or 超- to pretty much anything and as long as an obvious meaning is apparent, most people will go with it. Some, like -的 and -中, attach so easily that you might indeed question whether the results are "words" as such. If you wanted to argue that many compounds created spontaneously with -的 weren't "real words", and you accepted that this means that words like "lion-like" in English also aren't "real words", then I won't disagree with you. (I might have doubts about the usefulness of that definition of "word", but that's a different topic.)
However! Kanji that can be freely recombined like this are very much in the minority. You are free to invent any word you like, of course, but that doesn't mean that you can expect people to understand it. English is pretty flexible, too, but I still can't go around saying "Blueplate the whitenings before you exroom later," even if it would be obvious what I meant from context.
So whether you call clusters of kanji "words" or "combinations", some clusters are going to be more acceptable than others, and people do have to learn those clusters by rote. It's true that you can often figure out the meaning from the kanji, so that if you haven't encountered a word before you can make a good guess at what it means -- but it doesn't work the other way round. If you have a concept you want to express, and you don't know the word for it, you can't just throw kanji together and expect the result to be accepted without comment. You have to say it in other words that you already know.
(It's the same in English: Even if you know the morphemes "tele-" and "vision", how are you supposed to know that that's how you say "television" instead of "electro-opticon" or something? You aren't; you can't. Knowing the morphemes helps you figure out what new words you encounter mean, but it doesn't help you figure out what combinations are actual words.)
And so, with the exception of easily-combining kanji like 新 and so on, you really do need to learn all the words you want to use, one by one. You can only reliably get away with guessing from first kanji principles when it comes to passive tasks, like reading. (And even then, every so often you will make a mistake, because the meaning of some multi-kanji compounds just isn't transparent.)
Another way I might ask is, do Japanese consider "words" and "combinations" as separate things?
I don't know if this question is really answerable... until this question I wouldn't have thought that any English speaker would reject "neopragmatism" as an independent word, so clearly speakers of the same language can disagree on seemingly basic concepts like this.
But in the vast majority of cases, if you make up some combination of kanji that is meaningful to you and ask a Japanese speaker "is this a word?", then unless it's in the dictionary (and they know it) they will say "no", and if you ask them the same question about a compound that you know to exist, they will almost certainly say "yes". If you press them on the "no", they might expand their answer to "well, I can see how you mean X by that compound; I guess it's a word now that you've coined it," but the point is, we don't have anywhere near absolute freedom to mix and match if we want to stay within the bounds of "standard Japanese" rather than "psychedelic poetry."
Edit: Dave sez in comments:
I think your very last statement that deviating from standard Japanese lands you right into psychedelic poetry goes off the mark. Languages, and the people that use them, invent new words and uses all the time, which is how languages evolve as a matter of course.
Let me address this because it is a fair point. "Psychedelic poetry" was an exaggeration, yeah. It's absolutely true that new words are invented as a matter of course in language. The question isn't "Can words be invented or not?", it's "How much invention, and of what sort, can one get away with?" And the answer is determined by both cultural and personal factors:
Cultural: So as we all know, from at least the late Edo period through to the early Showa period, Japan was importing a bunch of concepts from foreign, notably European sources. They didn't have words for these concepts, so they invented new ones, and they often did it with kanji compounds, e.g. 競争 for "contest", 自由 for "free", 彼女 for "her" (women weren't a new concept, of course, but the highly European style of pronoun use was). So back then, you could get away with quite a lot of invention, although of course your newly coined word would have a certain nuance of foreignness.
But, after WWII, the pendulum swung hugely towards importing foreign concepts using phonetic loans (in katakana). This is why we say コンピューター instead of 電脳 or whatever. And as a result of this swing, it became less normal to invent kanji compounds, and so new kanji compounds became more noticeable and "odd".
Sidenote 1: There was also a reaction against foreign influence (historically, more about the early Chinese influence) in the shape of the Kokugaku school. They often coined new words using native Japanese vocabulary to replace existing Sino-Japanese words, and these new words had a distinctly "Kokugaku" feel. So even the cultural factors are not absolute; they depend on specific circumstances.
Sidenote 2: You can compare this kanji-compound frenzy with the use of Latin and Greek vocabulary to invent new scientific and engineering terms in English, back when modern science was new and had to name all these new ideas it was discovering.
Personal: The other thing about the "getting away with" concept is that it will depend on how different you can handle being. If you (not you personally, Dave, the general "you") are okay with everyone thinking of you as someone who invents new words all the time, and you think that the possible negative effects of this (people thinking you're a weirdo for not speaking "normally", work clients mistaking your creativity for simple error, etc.) are outweighed by the benefits of being able to say awesome things, then you will be able to "get away with" a lot more than someone who doesn't want to stand out that way and therefore has to stick to words that already exist.
And this, too, is not absolute: it's very common for families, both English- and Japanese-speaking and presumably everywhere else, to have their own jargon: words they invented or modified that everyone in the family understands but that no-one would dream of using outside the family, or expect anyone outside the family to accept as a "real word" (even if the meaning is quite clear).
So I guess what I'm saying is, I don't disagree with you that new words are invented all the time and there is no absolute prohibition against it. It may even be true that Japanese is more flexible in this regard than English because kanji are perceived as "modular". But there are still many factors, often changing drastically over time, which condition the relative "social acceptability" of different kinds of word-coining.