Is it permissible to take a loan from a bank for marriage and to start a business, with the sister paying the resulting interest, while the borrower only repays the principal amount?

Borrowing with usury is absolutely forbidden, whether for marriage or for a project, except in a case of dire necessity where there is no alternative to usury to avert it. It is not permissible for you to take this loan, nor for your sister to take it on your behalf, because the first is a direct engagement in a usurious contract, and the second is an assistance to it. Allah Almighty says: "And cooperate in righteousness and piety, but do not cooperate in sin and aggression." And the Prophet, peace and blessings be upon him, cursed the one who consumes usury, the one who pays it, its scribe, and its two witnesses.