Is it obligatory for a son living with his family to pay an agreed-upon amount to his parents, and is he sinful if he does not pay, knowing the possibility of forbidden money being involved in living expenses?

If a person is capable of providing for himself, his father is not obligated to spend on him. Rather, he must fulfill his father's request to contribute to the expenses if asked. If the father is content to spend, there is no harm for the son to eat from his money. The son is not obligated to share expenses with his parents, but contributing is recommended and is among the best deeds. However, if the father's money is mixed between lawful (halal (definition)) and unlawful (haram (definition)) sources, eating from it is permissible with some dislike, and it is better to do without it if possible.