Yes, you can buy houses and hotels when it's not your turn.
If another player expresses interest in the houses/hotels and there is a building shortage, an auction is to be held by the banker (even if it is still your turn).
You would have to sell the hotel AND the four houses (at half price), and would have no houses.
No, you may not. If you can't / don't want to buy that property, it must be immediately auctioned to the other players.
Yes, you would play that square. If you land on an unowned property, you may buy that property. If you land on someone else's owned property, you must pay them rent. If you land on a Community Chest space, you should draw a Community Chest card.
Yes, the properties that each player owns are public information.
No, you must keep your money on the table in plain view.
No, you may not move houses or hotels. If you want, you can sell the houses/hotels (at half price) and then build them in a new location.