Do light waves precisely follow null geodesic paths in General Relativity?

For clarity I think is best to start with Minkowski spacetime.

The equation we are trying to solve to understand the radiation of a point particle is: $$\square A^{b}=j^{b}$$

with the gauge $\nabla_{a}A^{a}=0$ and $j^{b}$ is the current density.

The potential \begin{eqnarray} A^{b}(t,x)&=&\int G^{b}_{a}(t,x,t'x')j^{a}(t',x')dx'^{3}dt\\ &=&\int\delta_{a}^{b}\delta(t−t′−|x − x′|)∕|x − x′|j^{a}(t',x')dx'^{3}dt' \end{eqnarray}

where $G^{b}_{a}$ is the Green function with support in the past light cone. In fact, the potential $A^{b}(t,x)$ only depends in the single event $(t',x')$ in the past which is the intersection between the null cone from $(t,x)$ and the world line of the particle.

Now in curved spacetime the generalization \begin{eqnarray} A^{b}(t,x)&=&\int G^{b}_{a}(t,x;t'x')j^{a}(t,x)dV\\ &=&\int\delta_{a}^{b}\delta(\gamma(t,x,t'x'))∕\Gamma(x − x ′)|j^{a}(t',x')\sqrt(g)dx'^{3}dt' \end{eqnarray}

where $\gamma$ is the null geodesic between the two points $(t,x),(t',x')$ and $\Gamma$ is the distance with respect the induced metric of a suitable spacelike surface that contains $x,x'$ does not work.

In general curved spacetimes the retarded Green function would depend on the whole causal past cone and not only the past light cone. This dependence comes from the interaction with the curvature and is related with the extra terms that you point out that vanish for Minkowski.

Therefore the potential is not only defined by the information that travel along the null geodesics but depends on the whole past of the particle. Nevertheless, singularities of the field travel along null geodesics globally. This is the content of the propagation of singularity theorems for linear hyperbolic systems and is related with the geometric optics limit.

As you required rigorous analysis I will point you to some papers with appropriate calculations:

Section 1.4 of http://relativity.livingreviews.org/open?pubNo=lrr-2011-7&page=articlese1.html

http://arxiv.org/abs/1108.1825

http://arxiv.org/abs/gr-qc/0008047

Also notice that my answer is just about electromagnetism in curved spacetime. To talk about General Relativity we would need to solve also for the Einstein's Equations. The point particle will affect the metric as self force corrections to the background metric. These type of corrections are treated in depth in the first reference.