What is the expiation (fidyah) required from someone who had intercourse, cut their nails, and shaved their pubic hair while in a state of ihram, being ignorant of these prohibitions, and how and where is it to be performed?

Thank you for your initiative to reconcile with your mother. The fidyah (expiation) is due from whoever commits acts forbidden during ihram before exiting it. However, sexual intercourse before completing the sa'i of Umrah invalidates it, and requires its completion, performance of a make-up Umrah, and a damm (sacrifice). If the person obligated was ignorant of the ruling, then nothing is required of them, neither sin nor fidyah. This is the preferred view of Shaykh al-Islam Ibn Taymiyyah. However, if their ignorance was about the punishment and not the ruling itself, then it has no effect, and this is the view of Ibn Uthaymeen. So, if you were fundamentally ignorant of the ruling, then nothing is upon you. Otherwise, it is obligatory for you to complete the invalidated Umrah, perform a make-up Umrah, slaughter a damm, and repent. Additionally, there is a fidyah for each forbidden act committed after gaining knowledge (fasting three days, or feeding six poor people, or slaughtering a sheep). As for performing Hajj while in debt, it is valid if one hopes to repay the debt, but dealing with usurious banks is forbidden.