I would also like an answer about the random order on the moderating tab. I assume it has something to do with the order students registered for the course (and then were registered by the new quiz system). It seems reasonable to expect the list to be sortable alphabetically by last name (at least), and maybe alternatively by the order of the most recent attempts. It is clear that neither is the order actually used at the moment.
As for knowing whose quiz you are grading, I strongly recommend speedgrader. You will see the same quiz display you are referencing, but it will be displayed within the speedgrader frame with the student's name, and you can click from student to student using the speedgrader frame.
Just like with other assignments, you can get to speedgrader from the grades tab, either through the dropdown menu for the quiz, or from the right side menu that slides out when you click the arrow next to the grade for an individual student on that quiz.
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