How similarly are light and matter affected by gravity? For a path of travel in a shape that is straight before it curves, if the curvature of the path is attributable to gravitational force, what additional information (if any) would be capable of indicating whether the path were taken by an object with mass or if it were taken by a beam of light? How would the implied features of the scenario (e.g., spatial scale, gravitational force, etc.) differ if the path were that of light versus that of an object with mass?
 A: In general a spatial trajectory can give absolutely zero information about the gravitational field or the nature of the object being acted on. A good example is parallel pencil beams of light. It turns out that in this situation, the beams have zero deflection due to each other's gravitational fields.
General relativity doesn't even really have a well-defined concept of a spatial trajectory in general. The concept is basically a Newtonian one. To define such a thing, you would need to be able to specify a global frame of reference, which doesn't exist in GR. See How do frames of reference work in general relativity, and are they described by coordinate systems? 
What GR predicts is the trajectory of a particle through spacetime. Given this trajectory, it's trivial to tell if it's a ray of light or a material object, because the ray of light moves at $c$. Such trajectories are defined to be straight in GR, if the particle is low in mass and electrically neutral. They're called geodesics.
A: The graviton propagator in de Donder gauge in flat space is
$$ D_{\mu \nu, \rho \sigma}  = \frac{1}{2} \frac{\eta_{\mu \rho}\eta_{\nu \sigma} + \eta_{\mu \sigma}\eta_{\nu \rho} + \eta_{\mu \nu}\eta_{\rho \sigma}}{k^2 + i \epsilon} $$
Given $2$ sources with energy-momentum tensors $T^{\mu \nu}_{(1)}$ and $T^{\rho \sigma}_{(2)}$, the leading order scattering amplitude for the interaction between them is
$$ G \  T^{\mu \nu}_{(1)} D_{\mu \nu, \rho \sigma} T^{\rho \sigma}_{(2)}  = \frac{G}{2k^2} \left(2T^{\mu \nu}_{(1)} T_{(2)\mu \nu} \ - \ T_{(1)} T_{(2)}\right) \qquad -(1) $$
where $G$ is the gravitational constant coming from the $2$ matter-graviton vertices, and each such vertex has coupling $\sim \sqrt{G}$. $T_{(1)}$ means the trace $T^{\mu \nu}_{(1)} \eta_{\mu \nu}$.
The gravitational potential can then be computed by taking the Fourier transform of the above amplitude. However, it is sufficient to look at the amplitude and read off the different behaviors of light path bending and matter path bending. Choose the first energy-momentum source $T^{\mu \nu}_{(1)}$ to be that of the sun. Then you can take the second source $T^{\mu \nu}_{(2)}$ as earth for a massive system, or light. Now comes the key point: light aka photon, is the quantum field of Maxwell's theory $\mathcal{L}_{Maxwell} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu}$. The trace of the energy-momentum tensor coming from $\mathcal{L}_{Maxwell}$ is zero: $T_{(2)} =0.$ So the second term in $(1)$ vanishes for sun-light interaction, while it doesn't vanish for sun-earth interaction. I don't know how one could use this to set up an experiment and measure the difference in interactions, but I suppose it is described somewhere, and this is the root of the differences that arise.
Reference: A. Zee, QFT in a nutshell
A: You are asking what additional information (if any) would be capable of indicating whether the path were taken by an object with mass or if it were taken by a beam of light.
Now in this case you need speed measured locally as additional information. The reason you need local measurements is that if you measure the speed of light from a far away observer's view, you could get a speed less then c (Shappiro effect).
Now if the particle's speed (measured locally) is c, it was light. The only other particles traveling at speed c are gluons, but they are confined, or gravitons, and they are hypothetical.
Anything with zero rest mass travels at speed c in vacuum, when measured locally.
Any other object, when you measure it, travels at speed less then c, measured locally in vacuum, is a particle with rest mass.
