who determines the number of properties?

The number of properties is generally 1. You generally don't need to live in more than one house. If you're working in some kind of situation where that's absolutely necessary because you're staying in distant places for extended periods of time, you'd need some kind of special exception.

Once everyone (or pretty much everyone, barring people who desire to live as nomads) owns a house, we can talk about letting people own vacation homes.

what if a landlord buys up 1000 properties and rents them out below market rates, is that systematic harm?

If they're making a profit, yes. If they aren't making a profit, no, but they won't be doing it for long unless they're getting funding from somewhere.