Is the amount considered a debt owed by the deceased father or by the mother in whose name the house was registered, and does the ruling differ if the father did not know the type of illness that led to his death but bought a car for his daughter a month before his passing?

Registering the plot of land in the wife's name without the intention of transferring ownership does not make it a gift, for intentions are what matter. The debt owed by your father must be paid immediately from his estate, as the Prophet, peace and blessings be upon him, said: "The soul of a believer remains suspended by his debt until it is paid off." Completing the house's finishing touches is up to the decision of the heirs. The sickness of death (marad al-mawt) is that illness which, according to medical judgment, frequently leads to death. A gift made during this state is considered a bequest, and a bequest to an heir is not valid unless approved by the adult, sane heirs, based on the Prophet's saying, peace and blessings be upon him: "Indeed, Allah has given every rightful person his due, so there is no bequest for an heir." Therefore, the car your father bought for your sister during his death sickness cannot be hers exclusively unless approved by the adult, sane heirs.