5.) b. his playing the chords improved our performance.
As a possessive pronoun, "his" show that something belongs to someone, in this case, it indicates that "playing the chords" (the act of playing, or the act of performing) belongs to a male.
6.) Mr. Williams is the teacher a. who went to such great trouble to help the students.
"Who" is used when referring to the subject in the sentence, in this case, is "Mr. William". "Whom" would be incorrect because it is only used to refer to the object of a verb or preposition.
7.) Would Agnes like to swim at Holden Pond with Elli and b. me?
"I" is a subject pronoun, meaning that it is the one that performs the action in a sentence, and "Me" is an object pronoun, which is the one that receives the action in a sentence, and sometimes are used after prepositions.
In this case, "me" is the correct sentence because "Eli and me" aren't the subjects of the sentence (Agnes is), and furthermore it is followed by the preposition "with".
8.) What do you think about a. him being chosen as class president?
Again, "him" is an object pronoun and is the one receiving the action in the sentence and it is placed after a preposition "about". "His" wouldn't fit here because it is a possessive pronoun, which it's used to indicate possession.
9.) Introduce the guests, with a. whom you will be sharing ideas for the fundraiser.
In here, "whom" is suitable because there's a preposition before it, and as the rules indicate: Whom is used to refer to the object of a verb or preposition.