"Views regarding one's past and future existence are included in the "62 false beliefs""
A little context: I assume the 62's mentioned above are the same as found in DN 1 (Brahmajāla Sutta) translated as "The sixty-two kinds of wrong views" by Walshe (MN 102 also develops on these wrong views).
Now, "wrong view" is understood as either "not according to the damma" or simply "not in accordance with reality". In this sutta, we read the Buddha say:
There are, monks, some ascetics and Brahmins who are speculators about the past, having fixed views about the past, and who put forward various speculative theories about the past, in eighteen different ways [...]
And he proceeds explaining the many ways in which a person may mislead him/herself into concluding something false about the past. The first, for example, is this:
Here, monks, a certain ascetic or Brahmin has by means of effort, extortion, application, earnestness and right attention attained to such a state of mental concentration that he thereby recalls past existences -- one birth, two births, ... ten births, a hundred births, a thousand births [...] Thus he remembers various past lives, their conditions and details. And he says "the self and the world are eternal, barren like a mountain-peal, set firmly as a post. There beings rush round, circulate, pass away and re-arise, but this remains eternally. Why so? I have by means of effort, extortion, attained to such a state of mental concentration that I have thereby recalled various past existences...That is how I know the self and the world are eternal"
Above he is denouncing the fragile leap of conclusion: a person remembered many past lives [and contractions and expansions of the universe] and without identifying an origin, he/she concludes there is none, "so it is eternal". Now, the buddha, when confronted with this particular question, simply declares there is no discernible beginning (SN 15:3) -- something very different from "it is eternal" and "it is not eternal".
Another example of misleading conclusion:
Here, a certain ascetic or Brahmin is a logician, a reasoner. Hammering it out by reason, following his own line of thought, he argues: "The self and the world are eternal"
In these ways, he alerts about the delicate trap of drawing wrong conclusions based on what one believes to be careful examinations (either by some experience or by some abstract reasoning) but that weren't careful enough. Moreover, he illustrates flaws using as example contemporary theories (eternalism, annihilationism, and the spectrum in between) pushed by other sects.
With these things in mind, he did not claim rebirth or views/knowledge of past existence to be false. Quite the opposite, he declares himself to have the ability to recall his past existences (and often describes episodes from them). Finally, exploring this ability is also part of an often-repeated formula for arahantship (and arahants are also described developing this ability). In MN 39, after exposing the Fourth Jhana:
"With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, he directs and inclines it to knowledge of the recollection of past lives. He recollects his manifold past lives, i.e., one birth, two births, three births, [...], many aeons of cosmic contraction, many aeons of cosmic expansion, many aeons of cosmic contraction and expansion, [recollecting], 'There I had such a name, belonged to such a clan, had such an appearance
Furthermore, he used to announce the rebirth of disciples who have died -- though not of the ones who reached the supreme goal (SN 44.9).
"Views regarding the future of the Tathagata (after death) are in the 10 or 14 "unanswered questions""
Yes. But it is important to emphasize it is "unanswered"; not a negative or positive. Even if we put aside the fact that this is a question about a Buddha, it is hard to use this to develop a trust on either view.
"The Buddhist doctrine of "anatta" (there is no self?) and "anicca" (self is impermanent?) seem to me to be saying that, if (it is believed that) there is rebirth, that 'rebirth' is fairly meaningless, i.e. it is a rebirth of nothing in particular: why not just call it a "birth" instead of a rebirth?"
As far as I can tell, in the [pali] canon it is not said "there is no self". Rather, mostly we read instructions to observe several [conditioned] things in order to directly see "this [the thing] is not self". Additionally, we read declarations such as "All phenomena are non-self" ("sabbe dhammā anattā" -- AN 3.134).
An important point to consider is that it appears something very particular is meant by "self" which does not fit well with our common understanding of this word. In SN 22:59:
Bhikkhus, form is nonself. For if, bhikkhus, form were self, this form would not lead to affliction, and it would be possible to have it of form: 'Let my form be thus; let my form not be thus.'
Is what is impermanent, suffering, and subject to change fit to be regarded thus: 'This is mine, this I am, this is myself? -- No, venerable sir.
Often times, when a challenger debates with the buddha claiming a self/eternalist view, he is debunked with questions such as "if that is so, can you change your form?". Therefore, I guess we need to, first, understand what is meant by "self" from propositions extracted from such discourses.
As for "birth" instead of "rebirth", though it's not clear to our reasoning "what re-borns" (all things above considered), the minimum takeaway of the use of this word in the discourses is that it denotes continuity.
"If rebirth happens that seems difficult to prove by personal experience"
Difficult indeed. But the texts support the possibility of personally testify it.
"is it an article of faith, not something one can verify by direct experience? If so isn't that (faith instead of experience) unusual in Buddhist doctrine (isn't doctrine meant to be measurable against one's experience of the world)?"
I guess it depends on what you mean by faith. I find it difficult to recall important claims made by the buddha that weren't suppose to be verified by one's effort in this lifetime -- so, as far as my knowledge goes, yes it would be unusual. From the canon alone, I wouldn't say this is a matter of faith in the devotional or "Belief" way, where one just take it as it is and assign a special value to it, often developing an emotional dependency towards it -- the clinging buddha frequently alerted about.
But faith as in a trust we give to something or someone while we are applying ourselves to personally verify it, I think it fits pretty well with everything else.
"Or if it is experience, what kind of experience (of other lives) is it, how are you supposed to know that so-called experience is not just a dream?"
I honestly don't know how I would proceed to scrutinize an experience of recalling my past lives.
"is it OK to believe, is it OK to say that a belief in rebirth isn't important to Buddhism? Not a big part of the historical Buddha's teaching? That when he mentioned it at all, it was to say that it didn't exist ("anatta" and "anicca"), that he didn't expect to be doing it himself, and that it wasn't worth talking about?"
One thing we can see explicitly in the texts is that he discouraged speculations, and this subject draws speculations easily, giving its such a puzzle for reasoning and concerns the mystery of our future. Having said that, I think its safe to say it wouldn't make much sense for him to require listeners to believe and internalize "the fact of rebirth", to consider it something central to the training (if it was that important, he could have made an extra noble truth out of it :).
There are a lot of aspects of the Dhamma that receive special attention and frequent emphasis. Urging people to believe rebirth as some sort of requirement or at face value is not something I've seen (MN 60 illustrates how the buddha approached this specific matter).
Now to the specific question: from the canon alone, rebirth is important (though, arguably not an urgent concern in the most practical sense). And, to say the least, the Buddha apparently considered worth talking about, so widespread this subject appears in the suttas, fulfilling roles therein, either as an aspect of the Dhamma, providing a broader exposition of our condition, or simply used to inspire and help us take the road to do something about it, etc. In the case of Dhamma, and this is significant, "no rebirth" does feature in wrong view.
But again, one thing is saying rebirth is important in buddhism. Other is saying (the belief of) rebirth is important to the practice. Other is saying rebirth is important to some buddhists ...
Edited to include a few more references (thank you @Unrul3r) and provide more direct answers to the questions