It is strongly implied, to the point of almost being stated outright, that Ming had been sexually molesting his daughter Aura—in public, no less—for years prior to the beginning of the film. When Ming uses his magic ring to force an orgasm on Dale in the throne room when Dale, Flash, and Zarkov first arrive, he asks Klytus, the head of the secret police, if he had ever seen such "responsiveness," to which Klytus responds "No, truly. She even rivals your daughter." It is thus implicit that Ming had been doing the same thing to Aura, presumably in front of his entire court, on a regular basis for years. Now consider Aura's personality, specifically, her hypersexuality, and her willingness to use sex to manipulate others. Those are both textbook responses to sexual abuse (n.b.: certainly not the only textbook responses, but still somewhat common).
Or, Klytus has been sexually active with Aura (either with or without Ming's consent). Sexual manipulation is commonplace in a Deadly Decadent Court; Aura probably traded sexual favors for influence.
Possibly, but his actions regarding her seem more like those of a frustrated suitor. That is, he seems to want her in no small part because theretofore he had been unable to have her.
Klytus had her. Aura says he wanted her all for himself.