Is the money added to my sister's bank account, which the bank describes as profits from an Islamic bank that has both interest-bearing and non-interest-bearing accounts, considered usurious money? If so, what are its rightful channels of expenditure, and is it permissible to give it to the poor?

The ruling depends on the basis by which the bank deposits the additional money into the sister's account. If it is profit resulting from a legitimate Mudarabah (profit-sharing agreement), then there is no problem with it. However, if the bank is not regulated and the matter is a loan, then the additional money is usurious interest that is not permissible for the sister. She must donate it and spend it on the welfare of Muslims or the poor and needy, while promptly withdrawing her money from this bank.