Not true. I'm a published, professional writer with an MA from a writing-intensive university (and a native speaker). I can assure you that native speakers still use this construction. It's odd and flowery-sounding, but definitely still in use in daily life. English is kind of all about variety. shrug
It's stilted, but still perfectly correct. I know people who say things this way, and I use this construction myself, but (with the possible exception of a few local variants) most people who use this phrasing are being deliberately poetic or flowery.
I hear it enough, though, that I think calling it archaic is going too far. "What saith the fox" (mentioned below) is genuinely archaic, but "what says the fox" is still correct. It's just quirky.
If Duo chooses not to mark it as "right", rest assured, that's just because it's odd, not wrong.
It's not archaic. It's not even obsolescent. It's alive and kicking all over the place. Possibly poetic, possible arch, possibly lofty. But alive.
How goes the war on terror? (John Arquilla, Leeds University 2014) How goes the war? (Clifford May, April 2003) "Well, Hardy, how goes the battle? How goes the day with us?" (Admiral Lord Nelson - Battle of Trafalgar 1805) "How goes it?" (Idiomatic, widely used, somewhat quirky = "How are you?")
The verb say (abair) is one of the 11 irregular verbs in Irish. Unfortunately you have to learn these verbs off by heart as they don't follow the standard rules. Here are all its forms: http://www.teanglann.ie/en/gram/abair
Inis means tell rather than say. It is a regular verb so the standard rules apply.
'Cad a deireann ...' was accepted as correct in another exercise. 'Deireann' is apparently the present habitual tense, whereas 'deir' is the present (simple). However, the distinction isn't observed by every speaker, apparently. http://www.daltai.com/discus/messages/12465/13182.html?1101981187
I went back to the Lenition lesson and this was the only mention of "deir" I could find:
'7. Other Words
Lenition is also used after the phrase nuair a when, the prefixes ró- too and an- very, and the word má if (unless the next word is a version of tá or deir). Other special cases will be highlighted in other lessons.'
"Cad a deir an sionnach?" doesn't really fall under this rule since "ma" isn't used, but because "deir" isn't lenited here even with "ma" absent, I'm guessing it may fall under the other 'special cases' the text mentions.
It's easy to give the rule, not to easy to say why. As you know, "a" is the relative pronoun. The rule is that a question word (cad, conas, cén) is followed by a relative clause. You're asking, as it were, "What is it that the fox says? " "Cé a bhuaigh?" "Who is it that won" - "who won?". For more, see: http://nualeargais.ie/gnag/satz4.htm