And If a woman and a goy man get married and then no longer live together, but do not want to give the divorce to each other, and are already living with other people, is this allowed, In the case the woman is living with another man, and the man living with another woman even though the 2 being married by the law of the country that would be allowed?
Regarding the law of the land, that you have to ask the lawyers, what is permitted regarding the local laws.
Regarding Halacha though, I am assuming that you are asking about two gentiles that got married. According to halacha there is no formal divorce process by gentiles, and when they are no longer living with each other and went their own separate ways they are no longer married. If you are referring to a Jewess that unfortunately married a gentile, she shouldn’t have married him in the first place, and as far as halacha is concerned their marriage was never valid, however since she had a reationship with a gentile she may not longer allowed to marry a kohen.
Rambam Hilchos Melachim 9-8.