If I add audio to a Pages document, export it as Word, then upload it to Dropbox, will this work?

Try it and then report back. You have the test environment already there.

If you try this with Pages v5.5.3 on El Capitan, the exported Word document with embedded audio (aiff) will generate a folder that contains the word document, and the separate audio file on the local filesystem. If you try this on a non-HFS+ filesystem that is not under control of the Apple Finder, it may bork, or not go as planned.