Unfortunately I only ever had the trial version of Sharepoint 2007 here so
many not be able to help, but...
a. I wouldn't really expect Word 2003 to deal with this /well/ because the
features that enable the links to Sharepoint data are only in Word 2007.
That means a conversion to Word 2003 format is going to lose those links or
transform them so they won't work the same way. If the conversion is
sensible enough to try to insert the necessary data into the Word 2003
document and throw away any link informaiton, at least you get to see the
current state of the document. But if it tries harder, then any link
information it tries to retain is likely to be lost rather easily.
b. when you get the the point where you make the copy, are you copying the
original Word 2007 document, or are you copying a Word 2003 document (.doc
format or old Word 2003 WordProcessingML .xml format) that you created by
(for example) opening the WOrd 2007 .docx in Word 2003 (via the converter,
obviously), then saving it as .doc/.dotx?
c. When the document is open in Word 2003, do the quick parts display as
field codes if you use Alt-f9 to toggle field code display (e.g. are they
actually INCLUDETEXTs, LINKS, or some such?

Peter Jamieson