Please let us know what you find out.
I tested this on our dev site and it works!!! I can't thank you enough. Our question pool was getting very murky, and this seems to be the solution. I now have to recreate in the Prod site, but I am so very grateful for your instructions.
1. I created site-level question bank with several question categories for the questions.
2. Then I created a few sample quizzes in a course.
3. I backed up that course and restored to new course.
4. I backedup up the restored course and restored again across multiple course categories.
And all looks perfect. The important lessons for me: site level questions, and question categories vs. course categories. With the new configuration, all questions stay in that one location even when the a course is restored within categories across the site. So far, all my questions exist only once, and are no longer multiplying and causing confusion. Again, my deepest thanks.