I would like those capitalists to show me the "property rights" molecule.
In feudalism, land ownership implies the obligation to protect the inhabitants and gives them certain rights. One remnant of this is that, in Britain, you have the right to walk on the undeveloped private property of others. Also, most European countries afford renters much stronger rights than the US does.
Many nomadic societies, from the Mongols to native Americans, don't recognize the right to own real estate you're not currently using. Sharia law has the concept that land can be unowned by anyone, and can revert to that state if unused, although many Islamic countries do not do that in practice.
So no, property rights do not stem from "natural law".
The people who believe property rights are "natural" aren't saying they aren't a human creation at all. Just that they are a human right, like other rights that are also human creations.
The justification for this is something like you own your labor and slavery is wrong, and if you use your labor to produce something, it's unfair for anyone to take it from you. Likewise it's your right to trade it for something someone else made, or do whatever. And from that you get capitalism and property rights as we know them.
However this argument doesn't cover property rights over natural resources or land. There are some arguments for that like the homesteading principle. But in general there is an argument to be made that natural resources should be publicly owned, or at least rented with the profits going to the public.
Also this argument is deontological. A consequentialist would say if violating someone's rights helps more people than it hurts, you should do it anyway. And an economist would point out that pure property rights don't work in the first place, because of market failures like public goods, common goods, information asymmetry, natural monopolies, etc. So everyone is better off if we violate them at least a little.
I don't think this is true; why would it need the "natural" moniker in such case? Natural rights proponents claim that such rights are not invented, but an universal truth, either endowed by a god (like in the US Declaration of Independence) or derived from some natural property, such as human reason.
This is contrast with people who claim that all rights are subjective creations, not an innate property of nature nor a consequence of such, and therefore are not universal.
First made famous by Henry George, if anyone is interested.
An interesting use-case for cryptocurrencies/blockchain-like technology.
