Is it permissible to use money earned by one's father from styling the hair of women (in a way that is) forbidden (in Islam) for marriage and housing, knowing there is a need for marriage and fear of temptation?

As long as you are able to earn a living, it is not permissible for you to benefit from your father's ill-gotten money, unless you are compelled to do so due to lack of work. In that case, it is permissible for you to take only what is necessary, but then it becomes unlawful for you to benefit from it once you are no longer in need. Your marriage is not considered a necessity that permits you to benefit from ill-gotten money or from a house built with it. You must adhere to piety and patience.