Is it permissible to deal with an American bank that requires money to be deposited therein for a legitimate online project, knowing that the bank does not invest the deposited funds and does not offer savings accounts, and provides a free guarantee for the money up to $250,0 in case of theft?

If your transaction with the bank is a deposit that does not involve an interest-based transaction, then there is no harm in it. However, the majority of jurists hold that the depositee is not liable for the deposit except in cases of transgression, even if a guarantee was stipulated. The Kuwaiti Encyclopedia of Fiqh mentions that the condition of guaranteeing a deposit is voided if it perishes without transgression; this is because the depositee's possession is a trust. Ibn Qudamah also affirmed that the trustee is not liable for the deposit, even if the owner stipulated it upon him or said: "I guarantee it."