1. If you use Moodle 2.9, issue 1. is no longer an issue. This was changed by MDL-6340.
3. Is not how Moodle random questions are designed to work. I don't understand how this requirement relates to point 2. If the students can only do a quiz during certain sittings, doesn't that automatically force them to wait before re-attempting. Anyway, the only way I can think of to do this is for you to manually randomise it, and make separate quizzes: Quiz for Session 1, Quiz for session 2, etc. Of course if you do that, combining the scores later will be a pain.
However, this also seems to conflict with requirement 1. Suppose there are 5 available sessions. Suppose that Student A does attempts in session 1. Student B does attempts in sessions 2, Student C in session 3, Student D in session 4. Then in session 5 they all want to make their second attempt. One of the students is going to have to see a repeated question.
4. Can be done a better way. Edit the definition of the Role that is doing the grading, and remove the quiz/grading:viewstudentnames capability. That will prevent them from seeing names while grading. (Note, this is not secure, if the grader goes searching round the UI, they can probably work out which student gave which response. However, if they stick to the grading screens, it is really anonymous. You can make it more secure by removing other capabilities. E.g. a grader role with just mod/quiz:view and mod/quiz:grade may be what you want, but test it.)
5. You cannot do this in Moodle, but you can later download the results as a spreadsheet from Quiz administration -> Results -> Grades, and check for these questions there. (I think it would be safer for someone to be checking this for the whole quiz, not relying on each marker to do this one attempt at a time.)