"What does he eat?"
Translation:Cad a itheann sé?
"arguably somewhat idiomatic"???
Sometimes the word following "how" is a verb, sometimes it's an auxiliary verb, sometimes it's an adjective, sometimes it's an adverb.
cad is an interrogative pronoun. It's always a question word that means "what", though it is idiomatically translated as "how" in cad chuige? and "where" in cad as (because English idioms and Irish idioms don't always overlap).
Can you point to some of these "different ways syntactically and structurally" that cad is used that make it so different from "how", in your opinion?
But "how" means the same thing in all of those sentences and is used in the same way. It's always a question word that queries and functions adverbially. There's no real distinction in meaning or function, although the usage in "how are you" and "how many" is arguably somewhat idiomatic. But the meaning is perfectly clear from the ordinary sense of the word "how." (That doesn't mean that "how" can be successfully translated into any other language using the same word for each of those questions, but that in no way indicates that there isn't a real fundamental unity to the sense and usage of the word in English.) By contrast "cad" is apparently used in different ways syntactically and structurally in Irish in different types of questions, which is much more confusing. Of course, English has its own peculiarities and absurdities . . .
atá is the relative particle a joined to the verb tá. With other verbs (such as itheann in this exercise), the relative particle a and the verb aren't fused together.
Your "thanks if answered" comment suggests that you don't understand the existing comments. In extremely simplistic terms, the relative particle is "that" in "what is it that he eats?".