The sentence with correct pronoun-antecedent agreement is:
Mrs. Smith was excited to teach her class.
In this sentence, "her" appropriately refers to the singular antecedent "Mrs. Smith."
The other sentences contain pronoun-antecedent agreement errors, such as using "it" to refer to "raindrops" (plural), "he" to refer to "brothers" (plural), and "she" to refer to "twins" (plural).