What are the Sharia obligations towards one's brother's wife and children—who survived the earthquake and are still in hospitals—and is he responsible for them, knowing that they have other relatives?

If your brother's children are sufficiently cared for in the hospital, then your only obligation towards them is to maintain kinship ties, inquire about their well-being, and visit them whenever possible. Maintaining kinship ties is determined by custom. However, if they are in need of financial support, then that is an obligation upon every wealthy relative who is an heir to them. Your interactions with your brother's wife should be in accordance with Islamic guidelines, as she is considered any other non-mahram woman.