1. (After several minutes,) I finally located my car keys. The phrase in parenthesis is a "prepositional phrase" (option A). This prepositional phrase is formed by the preposition "after" plus the adjective "several" and the noun "minutes".
2. Mikayla's homeroom teacher, (Mr. Jones.) is proud of her. The phrase in parenthesis is an "appositive" (option C). The appositive "Mr. Jones" provides additional information about the noun "teacher".
3. Henry's desire (to win) is obvious. The phrase in parenthesis is an "infinitive" (option B). The infinitive phrase in this case is used as an adjective which modifies the word "desire".
4. (Talking) is not allowed in class. The phrase in parenthesis is a "gerund" (option B). A gerund phrase includes a word ending in "ing" and functions as a noun.
5. Josie's project was a failure, having not prepared properly. This sentence contains a "dangling modifier" (option B). A dangling modifier is a a type of misplaced modifier, but it does not have anything to modify in the sentence; that's why the sentence should be restated so as to include the word/s that the modifier intends to describe.