The phrases in the sentences are as follows
"an ideal setting in which to govern" (sentence 9) Prepositional phrase
"Many of the most common ideas about Arthur" (sentence 10) Noun phrase
"Associated with..." (sentence 11) Prepositional phrase
"leader of battles" (sentence 12) Nominative phrase
What are the phrases?
"An ideal setting in which to govern" (sentence 9) is a prepositional phrase functioning as an adverbial modifier of the verb "offered." It describes the characteristics of Camelot that made it suitable for governing.
"Many of the most common ideas about Arthur" (sentence 10) is a noun phrase functioning as the subject of the sentence. It specifies the focus of the sentence, which is the origin of popular conceptions of Arthur.