I'm pretty sure that is due to the luminal propagation of gravitation...
erm... I don't know what you're talking about... other than gravity is not spread by light (e.g. black holes.) Irregularities in the orbit of mercury were solved if time, as well as the three other dimensions, were bent by matter.
About the instruments saying you go faster than the speed of light... We are assuming that the rate of speed is measured perfectly by an accelerometer. You feel like you're going faster and faster because you are going slightly faster, but mostly because your time moves slower than a "stationary" (i.e. not accelerating) observer. That is why inter-galaxy travel is possible in a lifetime... You get so close to the speed of light that you do not age but a few seconds.
So, what about objects without matter?
General relativity says that matter bends spacetime, to make seemingly straight lines not actually straight lines. Therefore, anything that ever travels through that universe must take a straight (seemingly "bent") path through space and time... That is why GR gravity is so elegant: And object acted on by "gravity" is just going in a straight path and retaining its inertia. Kind of similar to the Teenage Mutant Ninja Turtles pushing the ground up in FET, only nothing needs to be strangely accelerating.
Here's what I said. Matter (objects with mass... not everything has mass) may be the concentrated curvature of spacetime, therefore gravity is expected (mass="gravity".) So, to answer your question, objects without matter would lawfully go straight, without causing a curve of its own.