Is zakat obligatory on money set aside to purchase a house whose price has not yet been fully paid, and how is it calculated for past years if zakat has not been paid on it before?

You must first repent to Allah for your shortcomings in seeking knowledge and for delaying Zakat (definition). Zakat is a debt upon you that does not lapse with time, for the Prophet, peace and blessings be upon him, said: "Allah's debt is more worthy of being paid." Zakat is obligatory on saved money because it is originally a growing asset. You must calculate the time when your money reached the nisab (definition) (approximately 85 grams of gold or 595 grams of silver), then pay 2.5% of the money you had at the end of each lunar year, for each year. If you are unable to ascertain with certainty, then make an earnest effort and pay what will bring you certainty or a strong presumption of having cleared your obligation.