What is the ruling on taking the increase on inherited money deposited in a usurious bank out of necessity? What is the ruling on using skin-lightening creams? What is the ruling on curling eyelashes with a specialized tool?

It is not permissible for a Muslim to deposit money in usurious banks or to take interest from them, except in cases of extreme necessity, because usury is forbidden by Islamic law. Usurious interests are to be spent on public welfare for Muslims or on those in need.

Based on this, money must be withdrawn from the usurious bank. If your need and your sister's need reach the level of necessity for which public funds are disbursed, then there is no objection to benefiting from usurious interests. Merely completing studies does not permit taking these funds.

It is not permissible for both of you to exclusively inherit all of your father's estate; rather, you are entitled to two-thirds if he has no other children besides you.

The use of moisturizing and skin-lightening creams is permissible.

"Eyelash perming," if it means trimming or shaving them, is a forbidden plucking (namṣ) according to Islamic law, and its perpetrator is cursed by the tongue of the Prophet, peace and blessings be upon him.