To what extent is the hadith "Whoever has a growth from unlawful [earnings], the Fire is more befitting for him" authentic? And is it considered unlawful and sinful for the wife to feed her children if the father works in usurious lending and refuses to stop, and is it permissible for the wife to seek a divorce in this case?

Eating what is unlawful (haram) is forbidden in Islam and is subject to severe warnings, as the Prophet, peace and blessings be upon him, said: "No flesh that has grown from unlawful earnings will enter Paradise; the Fire is more befitting for it." And he also said: "Every body that has grown from unlawful earnings, the Fire is more befitting for it."

Your husband's work in usurious lending is one of the major sins. If you have forbidden him from it, then there is no sin upon you.

As for his expenditure on you and your children, if he has lawful wealth mixed with unlawful, there is no harm in benefiting from it. However, if all of his wealth is from unlawful sources, then it should not be benefited from except in cases of extreme necessity.