I don't see a .doc file getting small enough to outsize a HTTP request inside of it, even if you used some funky compression, but I'm willing to hear otherwise.
One question would be if you could upload the document once and then somehow trigger a very tiny edit that causes them to rescan it.
We do use LibreOffice to render previews of Office documents for viewing in a browser, and have permitted external resource loading to make those previews as accurate as possible. While this could theoretically be used for DDoS, we haven’t seen any such behavior. However, just to be extra cautious we’ve temporarily disabled external resource loading while we explore alternatives.
It may be that you are big enough that even the limited bandwidth you need for normal operations is enough to take out smaller hosts, so you'd need to measure and monitor to see how well this works.
Could Dropbox perhaps let me disable this feature? I almost never use the web interface so I wouldn't miss it and I prefer that my documents are not opened after being synched.
That does seem likely - dropbox tries to only upload diffs, when a file gets changed: https://www.dropbox.com/help/8/en