C) This Sunday, Shirley and me are going to go white-water rafting on the Chattahoochee River.
The pronoun "me" is wrong in this sentence. "Me" is an object pronoun meaning that it receives the action of the verb. "Me" does not do the action; it can not be used as a subject. In this sentence, the verb phrase is "are going". The two people doing the going should say "Shirley and I". A way to check to see if it should be me or I is to just say the pronoun and then the verb phrase. "I am going". Yes - this is correct. "Me am going". No - this is not correct.
Option A is correct. "You" is a subject pronoun, "him" is an object pronoun", and "his" is a possessive pronoun. They are all used correctly.
Option B is correct. "I headed out". Yes. This is correct. "I" is a subject pronoun.
Option D is correct. "You" is a subject pronoun. "Me" is an object pronoun, as in "You allow me." Not "You allow I".