Is it correct if I write "Já černou kočku vidím."? It didn't accept.
It is, although the meaning is "I DO see a black cat." "I SEE a black cat.", it is not a neutral information that you happen to see a black cat.
I will add it.
Why "Vidím černého kocoura" is not correct?
I do not think we accept kocour for a cat anywhere. It is a male cat specifically.