Did I commit a sin against my sister by withdrawing money from the bank to help my eldest sister, and what do I owe my middle sister as compensation for the bank interest, and do my mother and eldest sister bear any sin in this regard?

The sin of the money deposited in a usurious bank falls upon the depositor and whoever approved it. The interest from the deposit is unlawful money that must be disposed of by spending it on the welfare of Muslims or by giving it to the poor and needy. If the eldest sister is poor and in need, it is permissible for her to benefit from this interest.

The original amount that the eldest sister took from her younger and middle sisters is a debt that must be repaid to them. As for what the eldest sister spent on her sisters, if it was a donation and a gift, then she is not allowed to reclaim it from their money. However, if she spent with the intention of reclaiming it, she may deduct that amount from the debt. If she is able to repay the debt but does not do so, she is sinful.

Regarding the questioner's role in her middle sister lending to the eldest sister, she incurs no sin unless she deceived or misled her. Furthermore, she is not obligated to anything unless she guaranteed the repayment of the debt on behalf of her eldest sister.