Is it permissible for a Muslim man married to a non-Muslim woman to cover the funeral expenses of her deceased non-Muslim grandfather, knowing that there is no one else to pay, and otherwise his body will be cremated?

There is no objection to a man paying for the burial costs of a non-Muslim. This is an act of kindness to his wife and something he is commanded to do, performing it with the intention of winning her over to Islam. Islamic law has permitted the burial of a non-Muslim, even if he was a combatant, so the burial of a dhimmī is even more permissible. It is stated in the hadith: "Go and bury him," referring to the command given to Ali to bury his father, Abu Talib.