Is it also correct to say Livia non vir est? Or Livia vir non est? Or even Livia est non vir? I'm very confused by the word order.
Word order is rather fluid in Latin, though there are some general consistencies. For example, in general you will find the negation (non) directly before the verb (est). I have fairly high level of fluency in speaking Latin, and what sounds most natural to me is "Livia vir non est", though "Livia non est vir" is also acceptable. "Livia non vir est" may also work, though I believe that will be a different emphasis. You absolutely cannot say "Livia vir est non" - the non must come before the est. Additionally, if you list vir before Livia you change the meaning of the sentence (provided there are no surrounding sentences for context) to mean "The man is not Livia" (vir Livia non est) rather than "Livia is not a man" (Livia vir non est).