Is it permissible to befriend disbelievers, and if not, how is one to explain the Prophet's—may God’s peace and blessings be upon him—gift of a silk garment to Omar ibn al-Khattab, and Omar’s subsequent gift of it to his polytheistic brother?

Allah does not forbid kindness and justice towards non-combattant disbelievers. The prohibition is against allying with disbelievers, befriending them, and being pleased with their disbelief. As for loving a disbeliever due to kinship while hating their disbelief, there is no objection to it. Islam has permitted marriage to a woman of the People of the Book and enjoined kindness to disbelieving parents. The Prophet, peace and blessings be upon him, commanded Umar to give a piece of cloth as a gift to one of his relatives or sell it, so Umar gifted it to his polytheist brother from his mother's side.