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You're either very confused, or you have a much deeper understanding of physics than me. I love that it could be either. Does the conservation of momentum apply to things with no intrinsic mass?

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Yes. You can define a special-relativistic momentum and it's a conserved quantity, and it does have a nonzero value even for things with zero rest mass (which necessarily move at the speed of light).

(It may help to observe that an object with nonzero mass moving at the speed of light would have infinite momentum, and then it maybe makes sense that "zero times infinity" becomes a finite number)

Indeed. That's the idea behind things like solar sails, for instance. Check out the article here- http://en.wikipedia.org/wiki/Radiation_pressure and scroll down to "Radiation pressure by particle model: photons" if the top is confusing.

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