Is it permissible to give the obligatory Zakat on money saved in the bank to a poor sister whose rich husband does not provide for her? And is it permissible to use the interest from bank money to pay phone and electricity bills?

You may give your zakat to your sister if she is in need and her husband is not providing her with sufficient maintenance. You are not required to inform her that what you are giving is zakat. As for placing your money in an interest-based (usurious) bank, it is forbidden by Sharia. You must withdraw your money from it and place it in an Islamic bank. The usurious interest earned is not permissible to be used for paying bills or anything else; rather, you must dispose of it by spending it on public utilities, while seeking Allah's forgiveness and repenting.