The 'he' is hidden in the nincs. Nincs is a copulation of the words nem and van, and van is the third person singular form of 'to be'. So something is not well. Can be 'he', 'she', or 'it'. Of course you can also add the pronoun ő to make it clearer.

It is not okay. You have to use the adverb form "jól" ("well"). And with the adverb, you have to add a verb: "van". But then "nem" and "van" must come together and become "nincs" ("nincs" = "nem" + "van"). So, that is how we end up with "nincs jól".
Whenever "nem" and "van" come together, they must change to "nincs". They only stay separate if there is another word between them. You could actually say "nem jól van". It is a bit unusual but many people actually use that form.

Your sentence "Nem, ő nem jó" means "no, he/she is not (a) good (person)". In Hungarian the adjective and adverb forms of "jó" are clearly separated, you can't use one in the place of the other.