Is it permissible for a mature Muslim son to live off the expense of his unbelieving father and benefit from his money in legitimate ways? If that is not permissible, then is it permissible to work with a certificate obtained through studies financed by the unbelieving father?

Merely because a person is a disbeliever does not mean that his wealth is forbidden and cannot be utilized. The Prophet, peace and blessings be upon him, engaged in financial transactions with Jews and polytheists. He borrowed armor from Safwan ibn Umayya—who was a polytheist—on the Day of Hunayn, and he passed away while his shield was mortgaged to a Jew.

Therefore, if not all of the disbelieving father's wealth comes from usury or other sources forbidden by the Shari'a, there is no harm for the son in utilizing it for his expenses and studies. However, if his wealth is purely from forbidden sources, then it is not permissible for the son to utilize it except out of necessity. If it is mixed with lawful and unlawful sources, there is a difference of opinion. There is no objection to working with a certificate obtained while spending from the disbelieving father's wealth.