Is there a possibility, in either Moodle 1.9 or Moodle 2.3, to grade quizzes where the student name is not revealed to the grader?
There is no feature like this for quizzes in any version of Moodle. However Moodle 2.4 does support anonymous marking of assignment submissions.
Actually, this has been possible since Moodle 2.1.
There are two capabilities quiz/grading:viewstudentnames and quiz/grading:viewidnumber which control which aspects of the student's identity teachers can see while grading.
This is JUST what I'm looking for. But, even with the decribed permissions removed, teachers can select "Attempts" or "Grades" in the test and there the students picture and name is shown, and there they also can view their answers.
It is not shown when selecting "Manual grading", though.
Have I missed another setting?
Using Moodle 2.6.4
Right. This is not secure anonymous grading, that makes it impossible for the teacher to 'cheat'. It make it possible for the teacher to grade anonymously, if they work with the system.
View quiz reports
At least in Moodle 2.7 then the teacher can only access manual grading and statistics report.