"The apple is not black, but red."
Translation:Az alma nem fekete, hanem piros.
It's a grammatically correct sentence, but it doesn't make much sense. It means "The apple is not black, even though it's red."
If pedig stands before the subject or the object (note that the subject is implied here), it means even though
If pedig is placed after the subject/object of the second clause (which is not possible here), it expresses contrast, similarly to hanem, but the clauses don't exclude each other. "Ez az alma piros, az pedig zöld." (This apple is red, while that one is green.) Either both clauses are positive, or both are negative.
With hanem, one of the clauses is negated, because the statements can't both be true. "Ez az alma nem piros, hanem zöld." (This apple is not red, but green.)
Because "nincs" is to be used when the subject we are talking about is not there. In this case we are talking about a subject which is present, and we state that its quality is not X.
The apple is not here. - Az alma nincs itt.
There is no black spot on the apple. - Az almán nincs fekete folt.
As for "nem fekete van", you have to omit "van" because the verb is already implied. Check http://www.hungarianreference.com/Van-is-exists-omitting.aspx
After an adjective you don't have to use "van". So for example "Te piros vagy." (you are red) is correct, but in the third person singular you can't use the verb to be. "Nincs" means it's not/it isn't and it's the opposite of "van", but since you can't use "van", you can't use "nincs" either.