Is the moon terra nullius or do any property rights exist for it in any shape or form?
There are no property rights on celestial bodies ... at present
The Outer Space Treaty “explicitly forbids any government to claim a celestial resource such as the Moon or a planet.” And, since only governments can make such claims under international law, no one can.
However, the treaty was drafted at a time when only governments could play in space and didn’t contemplate actions by non-governments. The rocket fuel will hit the fan when a corporation starts exploiting space minerals.