Are the actions of the female pilgrim after Hajj correct? Does she have to assume a new state of ihram from the miqat upon returning to perform Tawaf al-Ifadah and Sa'i? Is it permissible for her to perform 'Umrah with her father, then the Tawaf and Sa'i of Hajj? What are the consequences of frequent argumentation during Hajj?

You made a mistake by entering the mosque while menstruating, and you must repent and seek forgiveness. What the Sheikh advised you is correct: you must return to Makkah to perform the Tawaf and Sa'i, as you are still in your state of ihram. You should go to Makkah without ihram and perform the Tawaf and Sa'i. Once you have finished the Sa'i, you will have completed the second release from ihram. It is not permissible for you to perform Umrah before completing the rites of Hajj. Imam Al-Shafi'i (may Allah have mercy on him) said: "If he performs Umrah while still in a remnant of his Hajj ihram... then he has no Umrah." As for the dispute, it suffices for you to repent, seek forgiveness, and increase in good deeds that expiate sins.