Look at animal society for an idea of what "rights" are natural. Now, that does not mean that different sets of rules aren't more or less useful than others. But that doesn't make them any less arbitrary. You can pick parameters that optimize the lift of a wing, but that doesn't make those parameters "natural" or "god-given." Whatever "truth" exists in those parameters is defined entirely by the desired outcome.

"Right to life" and "right to property" are socially useful fictions, encoding rules that optimize in some way desired social outcomes. They are useful only to the extent that they further those outcomes. The metes and bounds of those rights are defined socially. We might argue, internally, that some configuration of those rights would lead to outcomes that we as a society think are desirable, or that some configuration is incompatible with other configurations, but whether they are morally "right" or "wrong" is a socially constructed fact.

I'd only look to animals for ideas of where rights come from if I saw no distinction between the humans and animals.

There is evidence that suggests humans are born with moral intuition. There is a natural aversion to killing others (that can be overcome by conditioning) which suggests there is at least one natural law. There is also research suggesting that human babies have a sense for fair play and justice. I'll have to cite this at some point; I believe I read it a Malcom Gladwell book.

You also still haven't explained how society decides what is desirable and what is not. If you keep asking why, it will eventually lead to a natural right or fundamental good.

Aversion doesn't necessarily have to reflect an underlying natural law. In the case of killing, indiscriminate attempts to kill would threaten personal fitness, because there is a good chance the attacker could be killed in turn. This risk becomes more prevalent as tribes and societies develop to increase the fitness of their members. Indiscriminate killing within ones tribe would threaten group fitness, and so a viable groups members would need to kill the indiscriminate killer to increase chances of their own survival. Groups that do not act in this way suffer a big fitness penalty, and would not last very long.

Fit humans will not kill others of their tribe, but one merely has to look at the vast history of warfare to realize that there is no similar aversion or restriction on killing outside of one's tribe.