The fact that people believe in something does not make it real. I am not saying that people don't have belief. People believe all kinds of bullshit lies. It does not make them real. So just because a bunch of people think they are really girls when they are boys or the other way around does not make it real.
The idea of private property is not a social construct, even animals mark territory and defend it as their own. Society did not create that evolution and nature did. The idea of it is just that, an idea. It is not a fact or a truth. The fact and truth is there is only male and female, everything you have presented has not been based in facts.
Now you are arguing that social constructs are not real if they are just a belief that can not be proven. :/
What is it?
The belief is real. It is very important, because, true or false, these beliefs have real consequences.
Again, the belief or feeling is very real. As to whether it is true, your childish remark about chromosomes some comments above strongly suggests that you should learn very very much before you even start to assess this (did you even know that there are people who look like women, think of themselves as of women yet are genetically men?).
Some do, some do not. This does not mean that idea of private property is not a social construct, or that it is an evolved one. From what we know about so called primitive societies, their idea of private property, if at all existing, is very different from our western one. You should really do some learning.
The fact and truth is that you lack basic knowledge beyond high school level, not only in anthropology, but also in quite elementary biology and anatomy.