Does the prohibition of financial assistance include the wife if she did not sign it? And is it permissible for the husband to work in fields other than his studies when there is extreme need and to pay off debts?

Your husband must fulfill the conditions of the scholarship, and he is not permitted to work unless it is out of necessity, if the salary is insufficient for basic needs. As for your taking assistance, there is no blame upon you if the decision does not include you; otherwise, it is not permissible for you to take it.