Japanese is very flexible in such circumstances.
If you want to say " I don't read the newspaper " , you can say しんぶんをよみません or just よみません since the purpose of the question is to find out whether you read or not.
the question is probably なんがつうまれですか without the いま because it would not make sense
technically, replying with ごがつ is correct; however it is seen as a slightly rude reply so it may be better including です
Again, you can use whichever, as long as the purpose ( which is to find out what month you were born ) is stated.