"He is not well."
Yes, think of "nincs" as a closed formation of the words "nem van". To say "nem van" is incorrect, you have to say "nincs". You can only use "nem van" if you break it up and place other words in-between. Nem ott van" - "is not there". Etc.
One more important thing: "Nincs" is only used in the third person. With other persons you use the two-word phrase, because the word "van" needs to be conjugated:
- (Én) nem vagyok.
- (Te) nem vagy.
- (Ő) nincs.
- (Mi) nem vagyunk.
- (Ti) nem vagytok.
- (Ők) nincsenek.
"Nincs" itself cannot be conjugated, except for putting it in plural. Present tense, singular or plural, that is all that can be done with it. For everything else, the proper conjugated form of "nem van" has to be used.
You need a "van". The word "jól" is an adverb, not an adjective. So it needs a verb: "van".
So, when he is well: "Jól VAN."
When he is not well: "Nem van jól."
Except, when you have "nem van", you MUST use "nincs" instead! Hence: "Nincs jól."
End of story.
Some people might use a different word order and say:
"Nem jól van."
Let's call it a colloquial or regional usage. Not sure what category it falls into. But it could be acceptable even here.
"Ő nem jól." is incorrect, sorry. It is missing a predicate/verb. It means "He/she not well."
"Jól" is an adverb, it needs a verb.
The other two are fine. The word "nincs" is actually a verb, meaning "is not" in this case. "Ő" is optional, its use is up to context.
Some people will colloquially say this:
"(Ő) nemjól van."
"(Ő) nemjól érzi magát."
I made "nemjól" into one word, because that is kind of how it is used in this scenario. "Nemjól" as the opposite of "jól".