What is the ruling on relinquishing an inheritance to the deceased husband's family instead of claiming it, and does that absolve the husband of his outstanding debt? And what is the husband's standing before Allah if his brothers do not settle their share of his debt?

The debt of the deceased must be settled before distributing their inheritance. The heirs who benefit from the estate without settling the debt are in a difficult position. If the questioner has not received her share of the inheritance, she is not obligated to do anything towards her husband's creditors. The brothers of the deceased husband withholding her share of the inheritance is a clear injustice and oppression, because Allah the Almighty Himself took charge of the division of inheritances, and it is impermissible for anyone to prevent an heir from receiving their right. Allah the Almighty said: "These are the limits [set by] Allah, and whoever obeys Allah and His Messenger will be admitted by Him to gardens beneath which rivers flow, abiding eternally therein; and that is the great attainment. And whoever disobeys Allah and His Messenger and transgresses His limits – He will admit him to a Fire, abiding eternally therein, and he will have a humiliating punishment." The deferred dowry is included among the debts related to the husband's financial responsibility, so it must be paid from the inheritance before its distribution. There is no blame on the wife for demanding her deferred dowry and her share of her husband's estate.