Laws barring property rental are per se illegal, as the constitution does not give the government, at any level, the explicit right to dictate what one does (or does not do) with their own property. This goes for zoning as well.
Yes, it most certainly does. The Tenth Amendment states, "The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people."
Because there is nothing in the Constitution specifically prohibiting laws barring rental property, that power is reserved to the states or the people. So it is perfectly constitutional.
You know, I really wish you folks who are so obsessed with the constitutionality of things would actually read and understand the document. I don't blame you for posting AC; I would be embarrassed to put my name to that post too.