As tchrist's answer says, hypophora has made it into the Oxford English Dictionary, commonly seen as the most "authoritative" (in certain senses) dictionary of the English language.
Whether or not something "is a word" or "is a real word" isn't really answerable, so you'll have to make your own decision about its "status" as an English word.
As Laurel said in a comment, the OED only provides one citation (from 1656; the one in tchrist's answer) but the word has been used by multiple authors. Perhaps unsurprisingly, it seems to be easiest to find examples of this word in texts that are explicitly about rhetoric. As such, I think it could be described as rhetorical "jargon" or specialist vocabulary. Here are some 20th-century examples of this word that I found using Google Books:
HYPOPHORA : Greek ὑπὲρ, under ; φέρω, to carry ; is a form of speech by which the orator answers to his own question. It propounds an objection, and is the statement of the opponent's objection or of an argument which might be urged against the speaker's or writer's position. The Hypophora is followed by the answer or counter-argument called the Anthypophora. Hypophora is sometimes used as an equivalent to Eperotesis. It is a statement intended to forestall an objection or argument that will or may be used by the opponent.
(The Rhetorlogue: Or, Study of the Rhetor Or Orator, by John Demosthenes N. Ruffin; p. 290. The discussion of this rhetorical device continues to page 302.)
*Note: based on this definition, it doesn't seem to me that the word is really a good match to the phenomenon of "canonical questions" or self-answered questions on Stack Exchange.
Hypophora is one of the most effective means by which the course of a speech is interrupted and a lively conversational element introduced. Its most artistic form occurs when the orator brings forward views opposed to his own in the form of a question and then rejects them. It is a device closely related to prolepsis ; the only distinction being that in hypophora the objections are hypothetical, while in prolepsis a real argument is anticipated and overthrown.
(St. Augustine, the Orator: A Study of the Rhetorical Qualities of St. Augustine's Sermones Ad Populum, by M. Inviolata Barry; p. 142)
So far, I haven't found any examples of this word in English texts from before the OED's 1656 citation, but it does seem to have been used before then in Latin texts.
Also, hypophora does seem to originate from a valid Ancient Greek word (rather than having been put together from Ancient Greek word roots by some speaker of a modern language). Liddell, Scott and Jones's Greek-English Lexicon defines ὑποφορά as
A. carrying off below, purging, Hp.Coac.304, 511 (pl.), Heliod. ap. Orib.44.8.21.
2. draining off, Sor.1.58.
II. putting forward, by way of excuse, “ἡ τῶν μηνῶν ὑ.” X.HG5.1.29.
2. Rhet., = ὁ τοῦ ἐχθροῦ λόγος, Hermog.Inv.3.4, cf. 13 (pl.), Tib.Fig. 39.
III. drain, Gp.2.6.8, v. l. in Arr.Epict.1.29.40.
So I don't think it can be categorized as a neologism.