Is my Coral Beauty fighting with or trying to mate with my female Clown?

Many fish will do this, usually when one fish will fan their tail in the face of another, which seems to calm the more aggressive fish (if the more passive one is doing the fanning) or confuse the fish the aggressive fish is trying to show dominance over. Since both fish are about the same size and similar in aggression issues, likely it is just normal territory and aggression behavior and nothing to be worried about much.