Is what the questioner did—calculating the expenses for his deceased father’s treatment from the price of the car, which was in the father’s name and then transferred to the questioner’s ownership, and distributing the remaining price of the car among the heirs—considered Islamically permissible or not?

The car you registered in your name did not transfer ownership to you by sale due to the lack of explicit declaration. Its registration was to spare your father the difficulty of following up on the licensing. What you spent on your father's treatment was a duty upon you and is not considered a debt owed by him or the price of the car. Rather, the car must be included in the inheritance after his death. If the heirs agree for you to take it and compensate them, or if they waive their right, then there is no blame. Otherwise, it should be sold, and its price divided. Whoever spends on another without intending to be reimbursed has no right to reimbursement, and if the expenditure was a duty, then the intention of reimbursement is not valid in the first place.