A. I presume that you are talking about a perceived but imaginary image of a woman on the moon's face in the night sky, much as Americans perceive the Man in the Moon, and the Japanese perceive a Rabbit or Two in the Moon. Therefore, I would suggest 'in'. If, however, you are speaking of the first female astronaut to land there, then she would be the first lady on the Moon.
B. Your sentence is fine, but if you are narrating the legend, you might better use 'begin' than 'began'. It is impossible for me to judge from the context you have given.