Must I leave my younger siblings to their mother and her affairs, as my elder brother says: "Leave creation to the Creator," even though I bear full responsibility for the household and fear for them?

Being dutiful to your father after his death includes caring for your younger siblings and ensuring their proper upbringing and education in what benefits them. This is a form of maintaining kinship ties. A Muslim must teach his family matters of their religion and forbid them from unlawful acts. As for their mother, you have no guardianship over her, but you can advise her, enjoin good, and forbid evil with gentleness and wisdom.