@bbrogan , unless you are using some type of lock down/browser security when your students take their quizzes, then no, they can go in and out of the quiz as much as they want after they start it. That being said, if there is a time limit, the time will keep counting down from as soon as they start the quiz. If they exit the quiz the countdown clock will keep ticking. In addition, there is a feature called quiz logs that lets you see more of what students were doing while taking the quiz (for example that they left the quiz and came back, or if they opened up a different browser window) - How do I view a quiz log for a student?
As for messaging, I'm not sure what you're talking about. Are you talking about messaging in Canvas (chat or Inbox?) or a specific iPad feature/tool? Regardless, I'm not sure how you could turn off the feature (unless it's Chat in Canvas, then you would just hide it from the navigation in your course) and yes, as far as I'm aware the student could definitely use it during the quiz.
In regards to your question about keeping quizzes secure in Canvas, my recommendation is to check out this document - Quiz Settings to Maximize Security - and see if it helps.
My only other recommendation is to try and make sure that someone is constantly walking around the classroom/lab that the students are taking the quiz, to reduce the opportunity for cheating.
Hope this helps!
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