How did King Solomon and King David keep their marital duties to so many wives? Was there a system of rights the woman of the kings Harem enjoyed?
Once a man has fulfilled the mitzvah or peru irevu (having children) if the wife agrees to forgive his obligation then he's free from that obligation. S.A. E.H. 76:6 We know that Solomon had at least one son and two daughters so that would have qualified as having fulfilled the mitzvah, at least according to one opinion (bais hillel). We'll assume that he held like that opinion.
(Sorry, some sources are in Hebrew.)