Any force mediated by a massless vector boson will propogate at the speed of light.
It's not so clear that gravity is such a force: do we believe string theory's attempts to construct a theory with a graviton, despite string theory predicting a contra-factual massless scalar field, and still not being well-forumlated (is the background 10 dimensional? 11 dimensional?. . .), or do we take the fairly well-validated approach of general relativity in which gravity is the curvature of space-time, and not a field that propogates at all?
GR predicts the propagation of gravitational waves from mass/energy quadupoles (2 or more objects).
"string theory's attempts to construct a theory with a graviton, despite string theory predicting a contra-factual massless scalar field,"
The graviton naturally emerges as a spin 2 massless particle in any string theory. It is the closed string. The Hamiltonian for this closed string includes the momentum of the string which is a vector, so I don't see how it can be called a "scaler field".
"(is the background 10 dimensional? 11 dimensional?"
More like 26, since it takes that many to include chiral fermions. These theories though have duals. The 10/11 dimentional theories are equivalent to 3D 2t(time) theories and it depends whether deSitter, or anti-deSitter space is considered. The closed string in anti-deSitter space can exist off the branes and explains why gravity is so weak. Open strings are required to have their endpoints on a brane. In general it's better to use the simplest picture to conceptualize the phenomina.