Not to be argumentive but your statement of "a far more efficient method of managing the assessment process" I think is a little subjective. I administer a Moodle instance that has over 35k active users and 3k courses each semester. Our faculty is about 2k of those users. In my time as an administrator, I have got about half a dozen requests for something like you are asking; to be able to grade all of a student's assignments in one spot vs. all submissions for a single assignment in one spot. So your argument has been thought of by others and is valid. However, I think the vast majority of users prefer the method that is currently in place.
So to your goal, I don't think is possible at this time. You can sort by grade to get all the ungraded assignments together but you won't be able to get all ungraded assignments of a single student in one spot. As mentioned you can put in a Moodle Tracker feature request but that doesn't mean it will be done.
Finding unmarked quiz questions though is really easy though. When looking at quiz results, each question has its own column so you could sort by the column to get all the ungraded ones together; again that won't be by student but all the submissions together. If they are manually graded questions (i.e. essays), in the quiz submissions tab there is a manual grading method that makes it easy to find the ungraded questions.
Adding comments to assignments is probably best/easiest if the assignment was an Advanced Uploading of files assignment type. This assignment allows for response files. So if a student uploads an MS Word document, the teacher can download to their computer, use MS Word Track Changes for edits/comments, upload the response file and then the student can download the response file to see comments.
Happy 
oodling!