What is a "document of guardianship" (ṣakk al-wilāyah), and is it permissible for children to write it for their brother to be responsible for their father, who is afflicted with Alzheimer's, in receiving the inheritance?

Your brother has no right to be a guardian (wasi) or to manage your father's wealth or his share of his mother's inheritance. The appointment of a guardian is up to the Sharia court. Do not allow him to manage your father's wealth, and refer the matter to the Sharia court to consider placing your father under interdiction (hajr) and appointing a guardian over him if necessary. Your paternal aunts (ammat) have the right to receive their share of the inheritance, and your brother has no right of guardianship over them.