"According to the letter" is a prepositional phrase. PPs (prepositional phrases) have a preposition as head of the phrase. In this case, the preposition is "according to".
In C, "to win" is an infinitive phrase as it is formed by the infinitive particle -to and the verb in the bare infinitive form. What is more, "talking" in sentence 4 is a gerund. Gerunds are formed with verbs but act as nouns. As you can see, it is in subject position in the sentence.
I would say that "having not prepared properly" is a dangling modifier because it lacks the D.O (direct object) to refer back to Josie for the sentence to be grammatically correct. What was not prepare properly? The project lacks volition to prepare itself.