One thing to note is that with version 2.16, you can easily change a student's score on a problem while viewing that problem.
The current way to change a student's score is doable, but you need to know how. One way is to click Instructor Tools. Then select the student name and the problem set, and click "Edit one set for users". Scroll to the problem in question and you can change the "status" to 1 (for 100%).
You can submit an answer on behalf of a student, while acting as that student, if you change the permissions for doing that. In Course Config, in the Permissions tab, there is an item "Can submit answers for a student". You would need to change the permission level to professor to enable the ability to do that. I am unsure if the set's close date will still be a factor, and maybe if the set is closed this still might not let you submit an answer on their behalf.