I've been using the Stopwatch to give me an advantage in some harder areas, but I recently tried it against the Iron Golem boss and time didn't slow at all. Does the Stopwatch not affect bosses in general? Are there other points in the game when it does not work?
No, the stopwatch does not work on bosses.
From the StrategyWiki:
Another source is this guide on GameFaqs:
So the conclusion is that there are 2 events where it doesn't work on enemies: