I would argue that clocks don’t really run faster on the moon. What runs faster on the moon is time. The clocks tick at the exact same speed, it is just that time itself is running faster on the moon than it runs on earth. So from the earth’s perspective the clock is going faster when it really isn’t.
Yes. But to make up that difference, clocks on the moon would have to tick slower than clocks on earth to keep syncronized. Or they could do what you suggested, and add a leap second in every once and a while.