Yes, mostly, although it has to agree in number and gender. Inget/ingen (singular) and inga (plural). Do note, however, that whereas Dutch and German require geen/kein where you would have niet + een or nicht + ein, Swedish can (and will) sometimes express things with "inte en" and that's perfectly normal and natural.
It is coincidence. Frågar stems from Proto-Indo-European preḱ- (“to ask, woo”), whereas the Irish verb freagair ("to answer") is a prefix-verb combination of fris·gair, from fri (“against”) + gairid (“calls”), with the fri stemming from Proto-Indo-European *wert- (“to turn”). See https://en.wiktionary.org/wiki/Reconstruction:Proto-Germanic/fregnan%C4%85 and https://en.wiktionary.org/wiki/freagair
Nej is more of an interjection- it is just the word "no". Inga is used to negate har, so it's I have no questions, or I don't have any questions. Inga is specifically used to quantify things, so instead of just meaning no, it is saying that something is absent, or that something doesn't exist.
Inga is basically "Zero," as an amount (like one, two, zero, etc.). "I have no questions" = "I have zero questions."
Inte is "Not." As in, "I do not have questions."
Inte negates the verb, while Inga doesn't negate anything—it's just an "amount adjective" modifying frågor, and that "amount" happens to be zero.
It corresponds to kein in many cases but generally it is more similar to English no (as in 'no books', not as in yes or no, since we have a separate word for that, nej).
They use kein in many cases in German where we would use inte in Swedish instead (and you would use not in English).
"inta" is not a negation word. There's a verb inta, but that doesn't fit here (it means lit. "take in"). The word inte (which means 'not') only has one form.
For the difference between ingen/inget/inga and inte, that has been explained many times already in this thread, just scroll up.
"Inga" is the negating word here.
"Inte" means "not" but "inga" means "none [of something/anything]". Like in English, you wouldn't say "I have not none questions", you would say either "I have no questions" (Jag har inga frågor) or "I do not have questions" (Jag har inte någon frågor).
Hehe, thanks - I used to be annoyed by the many duplicate comments, but I realised some time ago that for many, a repeated answer can be worth a lot, even if it's been asked twenty times already. And I've also since learned that there's a bug in the Android app which will occasionally show zero comments regardless of how many are actually in the thread. I do generally refer to the answer instead of writing a new one, however. :)