What is the status of a Jewish couple that are married civilly but not religiously, meaning no Chuppah but the husband did give the wife a ring and said to her "you are Mekudeshet Sheli.." and now live together as husband and wife? Are they considered married under Jewish law and would the woman need to keep the laws of Niddah?

They are not considered married according to Jewish law. But because the husband was Mekaddesh (sanctified) the woman with a ring, and if there were two kosher witnesses the Kiddushin might apply, and even if he would not be Mekaddesh her with a ring, we still fear that they may be considered married because they are married civilly. So if they would like to separate, and the woman will wish to marry someone else, she will probably have to receive a Get (Jewish divorce certificate) L’chumra. But if they want to stay together, they should get married according to Jewish law. (אגרות משה אה"ע ח"א סימנים עד ועה, ח"ב סימן יט, ח"ג סימן כה, וח"ד סימנים עט ופא) In any case it is recommended to consult a Rabbi personally, because there are many details to clarify and these are very serious matters. Regarding the laws of Niddah, every woman even if she is not married, once she menstruates for the first time, she is considered a Niddah. And if they are together while she is Niddah they are committing a very severe sin (Rema Yoreh De’ah 183). So as long as they are not married according to Jewish law, if they keep the Laws of Niddah they will be avoiding the severe prohibition of Niddah, but will still be violating the prohibition of living together without marriage (Shulchan Aruch and Rema, Even Ha’ezer 26, 1). So it is highly recommended they should hurry up and get married according to Jewish law, and they will merit that the Divine spirit will be present in their home.