Yes, inte negates verbs and ingen/inget/inga negates nouns. Det finns inte … would be There is not … (negated verb) but Det finns inget/ingen/inga … means There is no … (negated noun).
inget is for singular neuter
ingen is for singular common gender
inga is for plural regardless of gender
It's like saying "Il n'est pas de café" in French or "Es ist kein Kaffee" in German: it doesn't make any sense! I can't give any examples in English because the verb in that circumstance is actually "is", but what I can say that is that it's an expression and that (I think) there is not real explanation for it.( even though we find the same thing in Norwegian: "det finnes")
Well, "exists" would imply that there exists no coffee whatsoever. That is not the point here. They have probably just run out of coffee for the moment. Therefore, "exists" would not work.
The word "finns" sometimes has the meaning of "exists", but you have to look at the context. Usually "exist" is not an optimal translation since we would use existerar instead of finns.