What is the status of a Cohen who marries a divorcee, or a ger? Does he lose his status as a Cohen, or can he still be called up to the Torah as a Cohen?

The man is still a cohen but he does not receive any of the honors of a cohen (one of them is being called first to the Torah) until he leaves the woman/women he is sinning with and accepts with a neder that he will not repeat the offense (Shulchan Aruch Orach Chayyim 128 40).