Is trading in restaurants at the time of Friday prayer forbidden by Islamic law, and is the money generated from it permissible, if the restaurant owner has gone to pray and left a woman to work in his place?

It is not permissible for your father to appoint a woman to sell things on Friday if her customers are among those upon whom Friday prayer is obligatory, because that constitutes aiding in sin. However, if she sells to those upon whom Friday prayer is not obligatory, such as women and travelers, then there is no harm in the appointment. If the sale occurs involving someone for whom Friday prayer is obligatory, it is valid according to the majority of scholars, with both parties being sinful according to the Hanafis and Shafi'is. It is invalidated according to the Malikis, and it is not valid for the one for whom Friday prayer is obligatory and is disliked for the other according to the Hanbalis.