Well, apparently not to Duolingo :) I took a stab at the translation without first "hovering" under the new word...and based on the closeness to freagra, I responded with "Who is answerable?" Duolingo marked it wrong (despite it meaning the same thing in English, let alone the literal translation, as you noted).
No, not really, unfortunately. Its meaning is broader than that. It's more like 'having the properties of, related to, or involved with something' For instance the word for 'Irish' (as in the nationality) is 'éireannach', which is derived from the genitive singular of 'Éire', and 'cóisireach' (meaning 'festive') is derive from 'cóisir' (meaning 'party').
It's a sensible conclusion to imagine that words in different languages will rarely coincide as exact synonyms. Why would freagrach automatically mean "responsible for" as well as "demonstrating responsibility"? In this case, it sort of does - depending on how you define your terms. The root of freagrach is freagair (to answer) so you would expect freagrach to include "accountable for/answerable for" in its meanings. If by "demonstrating responsibility" you mean "showing/accepting accountability" then, again, freagrach would be appropriate. But there is at least one other possibility. If by "demonstrating responsibility" you mean showing yourself to be serious/committed/sober/heedful/prudent or any one of a number of words in that register, then I would perhaps consider "stuama" or "cúramach". One cannot always be sure what DL is after, but one can be pretty sure that exact synonyms are as rare as hens' teeth.
I didn't assume that it would automatically fill every usage of the coinciding word in English, hence my asking the question. By "demonstrating responsibility" I meant responsible as a consistent adjective to describe a person, including things like getting work done in a timely manner, not procrastinating, consistency, being safety-conscious, and so forth.
When the same concept relies on the same word (freagra and "response") in both languages, it's reasonable to start from the premise that the concept is a shared concept, and that the usage of the term will be broadly the same. We're not talking about a single term being used for two different concepts in one language, and expecting that pattern to follow in the other language, just different aspects of a single concept.
Both languages have other words that might only cover one aspect of the concept ("blame"/locht, "reliable"/stuama), but that doesn't take from the fact that duine freagrach means "a responsible person" both in the sense of a person who caused something, and a person who can be relied on.
I'm thinking of two different variants on the meaning of "responsible" as the translation here, one of "Who is the responsible party in charge of what's going on here?" versus "who is a person who possesses the personality characteristic of responsibility?"
Which of those two variants would this "freagrach" sentence be referring to, or is it equally dually-versatile in Irish as "responsible" can be in English?
an bhfuil is the question form of tá (Bí is an irregular verb, that has a dependent form (fuil) that is used in the present tense for the interrogative and the negative). But Cé already has a hidden cupola in it, so you don't use an bhfuil with Cé.
atá is tá in a direct relative clause. It might be easier if you interpret Cé atá freagrach? as "Who is it that is responsible?" - atá meaning "that is".
Cé a rinne é? is "Who did it?", but again, it might be easier to understand what's going on if you interpret it as "Who is it that did it?"
It's all about the copula. It's easier to see it if you think about the answer:
Cé hé an múinteoir? - Is é Pól Pól an múinteoir or Is tusa an múinteoir
Cé atá freagrach? - Tá sé freagrach.
A statement like "He is responsible" doesn't use the copula ("responsible" isn't a noun form), so the question has to include the verb. A statement like "He is the teacher" does use the copula, so you don't need the verb in the question.