About point two:
Both supply and demand are incredibly constrained: building new houses is slow and not exactly being stimulated by the government, whereas demand is fixed because people need a place to live.
Property rental is an extremely attractive investment opportunity. Demand is guaranteed and due to the low cost of obtaining money it is basically risk-free. Even at near-zero rent you'll still make a profit - as long as the housing market doesn't crash.
On the other hand mortgages always carry some risk, and banks are aware of this. There are quite strict limitations on the amount you are allowed to borrow. This directly means that individuals will always be outbid by investors.
This directly results in price increases. Individuals are now no longer able to buy a home, forcing them to rent. Landlords, in turn, are happy to let you rent a property which you wouldn't be able to afford because it is way easier to evict you. And because demand is higher than supply, they'll happily increase the price. Rent prices are already in overdrive.
So now you're paying €1200 in rent instead of €800 in mortgage payment - for the same property.
Banning buy-to-let would result in a sale of the properties currently being rented out - which means a drop in the housing price. Rental supply will drop, but those houses don't magically disappear! People who previously were unable to get a mortgage can now suddenly afford to buy instead of rent. This lowers the demand for rentals, which decreases rent.
People don't need landlords - they need homes. With the exception of a very small number of short-term rentals, landlords provide zero value. Removing them from the equation decreases the prices for everyone.