No, you can't use Cmd + S for exporting a Pages document as a Word document. It is good that you save the Pages document in its native format first. If you tick a box after you have saved the document Pages thinks you have made a change and that is why you have the black dot again.
You should be able to assign a hot key to Export > Word. Mind that you don't use any hot key already in use. May I suggest you use somthing like Option + Control + s. The Control key isn't used much for hot keys on a Mac.
To do this follow the instructions here http://support.apple.com/kb/PH3957
If your important documents must be submitted in .doc or .docx format, and are being viewed and read by instructors assigning marks based on what they read (and on how it is presented), you might serously consider paying the price for the education version of MS Office (Mac).
Exporting of files requires the software to translate formatting between formats, and that translation is never perfect, For straightforward, text-only documents, you'll probably be OK. For anything more complicated than that, you'll start running into problems.