Not all second homes are private property necessarily. If you work out of it then it's personal property, like if you're using it as a vacation rental and doing all the cleaning and maintenance yourself. If you hire someone else to do the work for you then it becomes private property. My preferred way of explaining the distinction is that private property is akin to absentee ownership, while personal property is stuff that is in active use by you personally.

Yes, they are. Because by destroying the market, you also destroy the toothbrush making machines, and kill the toothbrush makers. Have fun eating the rich, but don't complain when they end up stuck between your theeth.