Are there any laws on how lands on the Moon are to be divided between multiple countries?
Space, including the Moon and Mars, is international territory. No country can lay claim to it, nor can any country use it to wage warfare. Countries also cannot place WMDs in space, though conventional weapons are allowed. All countries capable of reaching these bodies have signed and ratified the treaty.
I wouldn't say people are in a hurry to "conquer" the Moon and Mars. We're not even in a hurry to get there. Putting aside, for the moment, that all nations capable of reaching either the Moon or Mars have signed and ratified a treaty recognizing those areas as international territory, America is the only country to have ever placed people on a body outside the Earth, and we haven't even bothered doing it in over 40 years. Similarly, manned missions to Mars keep getting delayed and cancelled. The United States isn't planning on sending anybody to Mars until the 2030s.
It should also be noted that the people who have been sent to the Moon were not sent as part of a mission to conquer it; their mission included showing off in front of the Soviets and collecting scientific data. No bases on the moon were ever built and the last plan for a moon base was abandoned half a century ago.
In short, no country can claim land outside of Earth, which means that nobody can "conquer" it. Furthermore, few people have ever been sent to a celestial body, and nobody has in decades, despite the fact that we developed the technology to carry people there, so we can't say that countries are eager to send people there.