Is zakat obligatory on a sum of money deposited in a bank for the purpose of buying a house, and upon which a year has passed while it has reached the nisab? And is zakat obligatory on a small house that was bought for the purpose of living in it or benefiting from its increase in price?

Saved money designated for purchasing a home is subject to Zakat (definition) if a year has passed over it, and the obligatory amount is a quarter of a tenth (2.5%). As for a house you bought for dwelling, or for selling if its value increases, there is no Zakat on it, due to the lack of a firm intention for trade. The condition for Zakat on real estate is that the owner must have firmly intended it for trade.