Is it permissible to perform Tayammum for a month due to a medical condition that requires avoiding water?

Tayammum is permissible so long as the above excuse lasts. And Allah Knows Best.

I have been married for five years. My husband has two sons from his first marriage. Nevertheless, I took care of his children, and gave birth to a baby boy who is now three years old. Unfortunately, we keep fighting all the time; sometimes over his two sons and sometimes over his family. If I serve his family members and praise them, he treats me kindly. If I don`t, he turns my life into living hell. What should I do?

All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

Spouses should live with each other on a footing of kindness and equity. In your case, constructive dialogue is the best course of action. We advise you to be patient, commit your affair to Allah and ask Him to help you. We also recommend that you seek the help of the people of goodness and make them fix things between you and your husband. If all attempts for reconciliation fail then go to court, but we remind you that a problem, which could be solved with a beautiful word, is no problem at all, so praise family and children and try your best to win their hearts. And Allah The Almighty Knows Best.

Is it permissible for the big brother to force his sisters to wear Hijjab (Islamic wear) if their father didn`t do his part in this regard ?

All Perfect Praise is due to Allah, The Lord of The Worlds

The father`s attention should be drawn gently and politely, and the sisters should also be reminded with good words since Allah, The Almighty, Says (What means): "O ye who believe! Save yourselves and your families from a Fire whose fuel is Men and Stones." [Al-Tahreem/6]. And Allah Knows Best.

Does an internal examination require ghusl?

Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.