Laser related questions
1) Is it really as simple as grabbing a laser drive chip and a
  three-terminal diode from digikey and hooking them up according to the
  datasheet? Should the laser drive chip be able to handle all
  protection mechanisms necessary, or is there typically another device
  that's needed to handle some other form of protection?


The laser drive chips I'm familiar with are more about applying rapid modulation to the laser than providing DC power. Usually there's an additional power circuit required; and that power circuit is where the protection is normally implemented.

If you have a different type of drive chip in mind, please link the datasheet in your question.


  2) Is there a central regulatory body that does any testing to
  determine what class of laser you have, and whether your product
  follows all the necessary regulations?


In the US, it's up to the laser manufacturer to self-certify their product. You may be able to find a consultant to assist you with that process if you don't have the expertise.


  3) Are there any known issues using lasers with 1mm core plastic
  optical fiber? I know that POF has very different transmission windows
  vs. glass fiber, and I know that one of these optimal windows is
  650nm. 
  
  Would the beam stay narrow inside the fiber, or would it begin
  to disperse? 


The fiber is a waveguide, and the laser power will remain confined within the fiber core. It will attenuate (lose power over distance). There is also a process called dispersion which means different components of the laser power taking different amounts of time to traverse the fiber---but if you're not switching the signal quickly that's not likely to affect you.

Edit: A major difference between POF and glass fiber is that even in its transmission window, POF has much higher attenuation than glass. Attenuation in glass fiber is measured in tenths of dB per km. Attenuation in POF (last time I worked with it, several years ago) is measured in tenths or whole dB per meter.


  Would it still be coherent and collimated after going through, say, 15 meters > of POF?


The signal will still be coherent, but the dispersion effect I mentioned above may reduce the coherence length if you've gone through a very long fiber.

The output beam will diverge at a substantial angle (not strictly collimated) when it exits the fiber. The divergence is a diffraction effect and the angle is inversely related to the fiber core diameter --- meaning POF will have a lower divergence angle than smaller-core fibers. In multi-mode fiber like POF the output divergence angle also depends on details of the fiber construction. In general the output divergence angle will be similar to the input acceptance angle.


  I am investigating the laser approach, because it seems like most LEDs aren't even capable of 500 uW.


It doesn't matter much what most LEDs can do --- if you can find one LED that meets your needs, that is enough. And I think you should be able to find an LED to produce 1 mW and couple into POF, if you look long enough. But a laser should be able to do it more efficiently (but maybe more expensively). 

Edit: Be aware that using an LED does not reduce your safety concerns. 1 mW is still 1 mW and can still be dangerous. You will want the same safety precautions (you mentioned open-fiber control) whether you use a laser or LED. Regulations have not all kept up with the improved capabilities of LEDs in recent years, but that doesn't mean you shouldn't protect yourself and your users.