Can quick-drying mud be used on the same drywall joints where slow-drying mud was used?
In addition to the things @Steve Jackson mentioned in his comment, the quick-drying mud (called setting compound) is also much harder to sand than the slower drying-type compound, which makes it not such a great choice for your final coats where you're going to be doing the most sanding.  Pros use it because they're able to apply it so well that it requires little or no sanding, and it saves them a ton of time.  In my opinion, non-pros (including myself) do best to reserve it for those couple of joints that are wider than they should be and call for stronger compound that doesn't shrink.

Also, with drying compound the later coats will take less time to dry just because they're thinner, so it's not going to delay your project as much as you might think after waiting for that first thicker coat to dry.