The Torah gives us halachos of a king, but I'm curious whether anyone discusses whether it is halachically acceptable to have a female reigning monarch - a queen, and if there was one, whether she would fulfil any of the halachos of a king? I don't mean just as the king's wife, but as the reigning monarch.
It's unclear. We don't initially appoint woman to be the queen. The Sifrei says on the mitzvah of appointing a king says מֶלֶךְ וְלֹא מַלְכָּה. The mitzvah is to appoint a king and not a queen. However that is discussing the initial anointment. If the situation would be someone becoming a monarch as an inherited position it's possible that a woman would inherit her father's position.
The Minchas Chinuch on the Chinuch, Mitzva 497 raises and discusses the possibility that those who can not be appointed as king can still inherit the positions. He discusses the possibility of a queen being appointed under such circumstances but don't give a final ruling. Rav Schach in his comments on the Minchas Chinuch brings further sources to discuss the question but also indicates that he is uncertain what the halacha would be in such a case.
The Igros Moshe references that Minchas Chinuch in relation to a different question. He seems to lean towards saying that a queen would be appointed if she inherited the position (but that wasn't the question he was asked so he doesn't say definitively).