Commas should be used to set off the phrase "the lost princess," as it is in apposition to (gives a description of) the subject Perdita, but the sentence still makes sense without it. So, "Perdita the lost princess," contains the sentence's error, and the sentence should read, "Perdita, the lost princess, was found and raised by a shepherd."

In this sentence, "First" is used as an introductory word (specifically an adverbial phrase) and must therefore be followed by a comma: "First, take . . ."

The phrase "the medieval outlaw who stole from the rich and gave to the poor" is an appositive phrase. It restates the noun "Robin Hood" in a way that presents more details, but is not grammatically necessary to the sentence for the sentence to be complete and understandable. ("Most schoolchildren have heard the tale of Robin Hood" is a completely correct sentence on its own.) The appositive phrase should therefore be preceded by a comma ("Robin Hood, the medieval outlaw who stole from the rich and gave to the poor").

In this sentence, the phrase "a man known for his pugnacious character" is acting as an interrupting phrase. An interrupting phrase is a group of words which interrupts the flow of the sentence; interrupting phrases need to be separated from the rest of the sentence with commas (or dashes). In this case commas are needed after "Patton," and after "character."

The full, correct version of this sentence reads: "Patton, a man known for his pugnacious character, was renowned for his generalship."

Here, the phrase "in many great novels" is acting as an introductory phrase for the rest of the sentence. In light of the fact that the rest of the sentence can stand on its own (grammatically speaking), this introductory phrase should be separated from the rest of the sentence by a comma, which, in this case, it is. The sentence has no grammatical errors.

The best way to correct the underlined portion of the sentence is to use a single comma and separate it from the remainder of the sentence. The phrase "unsure of how to proceed" here acts as an introductory phrase, and it should be separated from the independent clause with a comma.

The full, correct version of this sentence reads: "Unsure of how to proceed, he hesitated at the top of the pass."

Here the phrase "one of the greatest generals of all time" acts as an introductory phrase for the rest of the sentence. The clause provides a relevant contextual detail, but one that precedes the grammatically essential structure of the sentence, therefore a comma is need between "time" and the rest of the sentence.

The full, correct version of the sentence reads: "One of the greatest generals of all time, Alexander the Great had a nearly perfect military career."

Here, the phrase "the biggest jerk in school" is an interrupting phrase in the sentence. The phrase comes in to give us a description of John's personality, but does not affect the grammatically relevant portion of the sentence, "I really hated losing to john, but he simply outplayed me." "The biggest jerk in school" should therefore be separated from the rest of the sentence by commas. Also, since the coordinating conjunction "but" is used, a comma is required after "school" in order to prevent a run-on sentence.

The full, correct version of the sentence reads: "I really hated losing to John, the biggest jerk in school, but he simply outplayed me."

"One of the most challenging art forms," is here used as an interrupting phrase. It should therefore be separated from the rest of the sentence by commas. The author's opinion that acting is "one of the most challenging art forms" is descriptive, but not grammatically relevant to the fundamental subject-verb relationship of this sentence. The sentence could read: "Acting is my passion." 

The full, correct version of this sentence reads, "Acting, one of the most challenging art forms, is my passion."

Here, the phrase "my favorite food" is being used as an interrupting phrase, therefore it must be separated from the rest of the sentence with commas. "Cheese," the subject of the sentence, must be connected to the primary verb "is," so the interrupting phrase must be enclosed with commas, one on either end.

The full, correct version of the sentence reads: "Cheese, my favorite food, is the main reason I can never become a vegan."