Is it permissible for the borrower to surrender his house to the bank for sale due to his inability to repay his debt and its interest, and then pay the remaining agreed-upon amount after the house is sold, in accordance with the lawyer's advice, or is he obligated to keep the house?

If you wish to keep the house, there is no harm in retaining it while paying the usurious interest to the bank. However, you are obligated to repent from usury. The rule is that whoever purchases something with a usurious loan owns it, while bearing the sin of usury. If a person cannot avoid paying usurious interest, it is permissible for him to pay it, along with repenting from the original prohibited transaction.

If you do not wish to keep the house, and you do not have money to pay what you owe after the bank sells the house at a loss, it is permissible to approach the National Debt Helpline so that they may assume the debt on your behalf, and you would repay them in installments, provided that they do not charge any increase beyond what they will pay to the bank. Otherwise, this would be prohibited usury, and in that case, you must keep the house and pay the bank, so as not to enter into a new usurious transaction.