A gerund is the ing form of a verb used as a noun. There is no common 3rd word. There is always a direct object. I have laid the jecket there every day. He looks like me.
None is singular, except when it is plural.
His examples persuade me because I am already persuaded. A school bus was involved in an accident. None of the children were hurt. Yes it is correct, and yes it looks like a double possessive. Both are correct.