Will Dropbox re-upload a changed file if I don't access it from within the app? I know that when I edit a file from within Dropbox (using an external application), after I close said file Dropbox uploads the updated version. Does this also work if, once I have accessed the file from within Dropbox once, I open the file from within the editor app (e.g. using a "recent files" menu?)

If this is not the case, I'd be looking for a way to achieve this effect.
For Dropbox and OfficeSuite (if you edit from the recent files), they seem to sync.

I'm not sure with other apps. In any case, if you need a two-way sync, you can always use DropSpace. See this similar question: How do I sync with DropBox?