What is the Shar'i ruling on mortgaging one's house to the bank to help a brother obtain an interest-based loan? Is it permissible to accept a gift offered in return for that? And should Hajj be postponed in case of such assistance until the loan is repaid?

It is not permissible to assist a brother in obtaining a usurious loan, whether by taking the loan for him, guaranteeing him, or mortgaging anything for his sake. This is because it falls under cooperating in sin and aggression, which is forbidden by Islamic law. He must be advised to fear Allah and abandon usury. The husband must hasten to perform Hajj if he is able.