Are these seven sentences correct and natural?
1. The importance of the test will sometimes (intensify / heighten / aggravate) the nervousness of the students.
2. Any opposition to the rules is unbearable.
3. The stolen property was (restored / revitalized) to its owner.
4. He truly excelled among all the football players. (I want to say that he's head and shoulders above the others, can I say such thing using the verb "excel"?)
5. The advertisements should not be misleading.
6. They were resolute to go to graduate school.
7. Canada (appealed / petitioned) the United Nation to consider its proposal.

Question

Are these seven sentences correct and natural?
1. The importance of the test will sometimes (intensify / heighten / aggravate) the nervousness of the students.
2. Any opposition to the rules is unbearable. 
3. The stolen property was (restored / revitalized) to its owner.
4. He truly excelled among all the football players. (I want to say that he's head and shoulders above the others, can I say such thing using the verb "excel"?)
5. The advertisements should not be misleading.
6. They were resolute to go to graduate school.
7. Canada (appealed / petitioned) the United Nation to consider its proposal.

Answer

These are more natural:

2. Any opposition to the rules is discouraged.
3. The stolen property was returned to its owner.
6. They were determined to go to graduate school.

Answer

1) intensify or heighten work
2) I think "unacceptable" or "intolerable" would work better.
3) "returned" in this context: to give back to the owner. If instead you meant to make it like its original conditions, then you would use the two words you have and change the owner part to refer to condition.
4) it works
5) works
6)works, but the word is not commonly used in conversation. More for writing. But it does work and is correct.
7) "appealed to" but "petitioned" would work also