"It's" is a conjunction that stands in for the phrase it is. Anytime you intend to use the phrase, "it is" you may use the conjunction "it's" instead.

Use an apostrophe to indicate possession, if the noun ends in "s" then the apostrophe should follow the "s."  If the noun does not end in "s" then add an "s" after the apostrophe.  The best way to correct the underlined portion above is: "Paul's family was shocked by his confession in open court:"

The word "wont" means something entirely different from the word "won't," which is a contraction of the words "will" and "not."  The latter is clearly what should be in the sentence above, and as such, the best way to correct the underlined portion is: I am wont to be late to work on Mondays:"

"Wont" is a somewhat antiquated word, but it is still within the English language. 

“It’s” is a contraction meaning it is. “Its” indicates possession when using the third person pronoun it. The two are very often confused. The best way to correct the underlined portion of the sentence above is: "deprived of its courtesy protocols"

“It’s” is a contraction meaning it is. “Its” indicates possession when using the third person pronoun it. The best way to correct the underlined portion of the sentence above is: "It's unfortunate that you can't seem to behave yourself in class;"

Note that semicolon correctly joins these two independent clauses into a compound sentence. Another correct option would have been to add a comma and a coordinating conjunction (and, but, or, nor, for, yet, so) after class.

“It’s” is a contraction meaning it is. “Its” indicates possession when using the third person pronoun it. The best way to correct the underlined portion of the sentence above is: "The tank was so badly damaged that its treads were entirely nonfunctional."

The word "cant" means hypocritical or sanctimonious talk, which is entirely different from the word "can't," a contraction of the words "can" and "not." The former is clearly what should be in the sentence above, and as such, the best way to correct the underlined portion is: "The politicians vapid cant did little to appease voters:"

As a general tip, "can't" is in the present tense (the past is "couldn't"), while "did" is in the past tense, so even if you did not know the word "cant" you would still have been able to tell that the sentence is incorrect as written. Note also that a colon is correctly used to connect two independent clauses, where the second clarifies the first. A comma is an incorrect way to separate two independent clauses.

The word "wont" means habitual or customary behavior, and obviously is entirely different from the word "won't," which is a contraction of the words "will" and "not." The latter is clearly what should be in the sentence above, and as such, the best way to correct the underlined portion is: "The simple fact of the matter is, if you won't start treating people more kindly,"

It sounds like the addressee here is "wont" to treat people unkindly.

“It’s” is a contraction meaning it is. “Its” indicates possession when using the third person pronoun it. The best way to correct the underlined portion of the sentence above is: "It's extraordinarily difficult to make a living as a stage actor, but I feel it's entirely worthwhile to try."

“It’s” is a contraction meaning it is. “Its” indicates possession when using the third person pronoun it. The best way to correct the underlined portion of the sentence above is: "The robot is badly damaged; it's not fully functioning as a result of the horrible crash."

Note that a semicolon correctly separates the two independent clauses making up this compound sentence.