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2c:
Photons are massless. The path taken by the particle is affected by the curvature of spacetime, which is caused by mass/energy - the effect is called gravity - General Relativity is the theory that applies.
As to the box experiment, I have no idea, but perhaps it has to do with how the reference frames are constructed.
And then there's quantum gravity ........ too hard!
An additional thought:
Consider a black hole, from which no light can escape. It's not that the massless photon is "pulled on" by gravity, it's that the curvature of spacetime is so tight that any path the photon takes curves back inside the black hole. Alternatively, you could think of it as the speed of light being fixed in all inertial reference frames (Special Relativity) and the photon never achieves "escape velocity" from the black hole - escape velocity is independent of vehicle mass.
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