Should America disallow other countries from using space/moon

Firstly, America has no claim on space, which is pretty huge, and it has no claim on Earth's moon. Putting a flag up there is sort of like putting a flag on Mount Everest (it is traditional, when someone climbs Everest, that they leave a flag, or some kind of token to indicate that they successfully made the journey; the flag or token does not mean they own Mount Everest, obviously). The United States was not even the first country to explore some part of space. By your logic, if being first up gives one ownership, Russia should own space.

Secondly, I feel sorely obligated to point out to you that China is not a third world country. China is a first world country. In fact, China is significantly wealthier than America.

Thirdly, many countries have been operating in space for decades, ere on half a century. Perhaps you've heard of the International Space Station? It's called 'International' for a reason; sixteen countries, the U.S., Russia, Japan, Canada, Brazil, Austria, Belgium, Denmark, Finland, France, Germany, Great Britain, Ireland, Italy, the Netherlands, Norway, Spain, Sweden, and Switzerland, worked together to build that station, and it is internationally accessible. Launching anything into space is not considered an act of war, it is considered an act of exploration/science.

Fourthly, even if the US were to make such a foolhardy move as to declare space 'off limits' to the rest of the world, how would they enforce it? America is trillions of dollars in debt. It could not go declaring war on every country (especially not its allies, of which China is one) that went to space even if it wanted to. The United States would barely be able to stand against one superpower, let alone a coalition of the entire world, if it decided to go attacking everyone that wanted to visit their space stations and such. America has no authority over the rest of the world, aside from empty threat of retaliation. It certainly couldn't make that threat over the entire world.