Quizzes are saved into the database and the mod doesn't delete any database data on uninstall (if you want you have to do it manually), so you should not lose anything (make you can always make a backup just in case
Additionally the install already takes care of updating the database if needed from version to version, so you should not have problems.
BTW I uploaded another version of the package restoring the behaviour of not delete the quiz_images while uninstall (I removed it a while ago...honestly I don't know why...
On my test forum the menu bar says Quiz. On the main forum SMF Quiz.
That's curious...I see Quiz when on the forum and SMF Quiz when in the admin area.
If this is the problem then you can change:
$txt['SMFQuiz'] = 'SMF Quiz';