Is what a woman spends from her inheritance on her household and children considered zakat, or is she obligated to pay zakat? If obligated, is it due on the profit or on the principal amount?

Placing money in interest-based banks and taking interest is usury (riba), which is strictly forbidden and considered one of the major sins by consensus. One must repent from it and withdraw the money. It is not permissible to benefit from usurious interest; rather, it must be disposed of by spending it on charitable causes and for the poor. What was taken out of ignorance is forgiven. As for placing money in banks without interest, it is safer to avoid it unless there is a necessity, as it might be considered aiding in sin.

What a mother spends on her children is not counted as zakat (definition). If she is obligated to spend on them, they become self-sufficient through her spending, and it is not permissible to give them from zakat.