1. The professor assured Mark and I that we could still pass the course.
Congratulations, you answered correctly. The answer is NO. The sentence is grammatically incorrect.
The sentence should read: "The professor assured Mark and me that we could still pass the course."
The subject of the sentence is "professor". The verb is "assured." "Mark" and "I" are objects. They are the ones being assured. They are not the ones doing the assuring.
Therefore, "me" (object) is correct, not "I" (subject).
Congratulations, you answered correctly. The answer is NO. The sentence is grammatically incorrect.