Is the questioner sinful for not informing his grandmother that he took a portion of the money from the sale of her house to buy a car and furniture for the house, and is he absolved of responsibility by secretly giving her money?

If the grandmother is discerning and of sound mind, it is not permissible to dispose of her property without her permission, for the Prophet, peace and blessings be upon him, said: "The property of a Muslim person is not lawful [to take] except with his good will." Therefore, your purchase of a car from her money without her permission is considered an تجاوز. You must repent and return the money to her immediately or seek her permission. If you fear harm to her, then do not inform her and return the money by any means. However, if she is not discerning, then her permission is not considered. As for her writing the house in the name of her daughter (your mother), this is not sufficient for her to exclusively own it, as a condition for a gift is that the gifted person takes possession of it during the lifetime of the giver, and possession is not achieved merely by writing.