Is it correct to say "Die Kochin isst das Pferd"?
Almost. "Die Köchin isst das Pferd" would be right. Since you can't tell the cook's gender from the English sentence, both are accepted.
Why is it "das Pferd"? Shouldn't it be "ein Pferd"? "Pferd" is an object here, not a subject.
das Pferd = the horse
ein Pferd = a horse