Doing a backup and restore of a course with a link to a Public Questionnaire doesn't behave as expected.
Create a public questionnaire in one course. In a second course, add a questionnaire activity that points to the original questionnaire with the "Use Public" option. Backup the second course and restore to a new course. A new copy of the original public questionnaire is created in the restored course, resulting in a duplicate public questionnaire. (The advanced settings tab, ability to view responses and change questions are available in the restored course for this new questionnaire. The course restored from did not have these options, since the questionnaire lived elsewhere.)
What I would expect would be that the restored course should link back to the original pubic questionnaire like it's source course did, rather than creating a second copy of the questionnaire.