Will the speed of gravity and the speed of light always be the same? I have been considering the following though experiment: If the entire sun were to disappear at once, would we feel it first due to the sudden drop in gravity or see it first? The answer I have come across is that we would feel and see it at the exact same time. I know though that light will travel slower through different medians. This means that the time it will take the light to reach us will actually be slightly greater than the distance divided by the speed of light. Will gravity also be slowed down or will we feel the effects of gravity dropping off slightly before we stop seeing the light?
 A: General relativity can be expanded in orders of $G$ and $1/c^2$, so that the lowest order term is Newtonian gravity, the next order term gives the correction to orbits such as Mercury around the sun, and the third order term accounts for the gravi-magnetic field detected by Gravity-B probe. These first three orders, sometimes called N, PN, PPN and PPPN, for N meaning Newton and P meaning post. This is the post Newtonian expansion for gravity. The next order term gives equations that are similar to Maxwell's equations. This is the domain of linearized and weak gravitational radiation. It describes gravitational waves that move at the speed of light.
This means the speed of gravity is the speed of light. This leads to the question of whether there are indices of refraction, or something similar to a change in the dielectric constant. A stationary gravity field induces Einstein lensing, and this is similar in a way to an index of refraction. A gravitational wave in a strong field limit might do much the same. In a funny way a strong gravitational wave could be thought of as a nonlinear index of refraction that propagates in a medium with an index of refraction. 
There is the Sach's peeling theorem that classifies vacuum solutions to the Einstein field equations in a way similar to the near and far fields in electromagnetism. The nearest field is the Type D solution for a black hole, which is like the direct source. This "peels off" with greater distance from type D solutions to type II type III solutions out to type N solutions for gravitational radiation. The type II and type III solutions are a bit strange, and these could in some physical sense be seen as gravitational radiation that is self-interacting or partially bound. This connects in some ways to the geon. 
The geon can be thought of as produced by imploding spherical gravitational waves. If this is too weak it scatters back out, but if strong enough it may implode into a black hole and scatter out some gravity waves. The geon is the idea of a self-bound affair that is neither a gravity wave or black hole. This appears to be unstable, a bit like an unstable fixed point at the top of a potential hill.
If you have a gravitational wave that is self-attracting and strong enough to partially bound itself it then has an effective mass. These have longitudinal modes as a result. Type N gravitational waves are purely transverse. In the case of type II or III solutions or the transient state of a geon the curvature of space is a sort of index of refraction that adjusts the velocity it propagates.
