Time evolution of the worldlines of 2 particles Suppose I have a lab frame that is freely falling in a gravitational field of the Earth -- assume non-homogeneity-- and a uniform constant electric field. There are 2 test particles in the frame -- both of mass $m$, but one is of charge $e$ and the other neutral. They are initially separated by by a vertical distance $h$. I would like to model how their distance evolves. Could anyone help me?

Things I've thought of (but may not be entirely right): I shall assume that the interaction between the particles is negligible. 
Then the geodesic equation for the neutral particle is $$u^a\nabla_a u^b=0$$ where $u^a$ is its 4-velocity. 
The worldline of a charged particle is 
$$u'^a\nabla_a u'^b=\frac{e}{m}F^b{}_au'^a$$
where $F^b{}_a$ is the electromagnetic tensor.
And then...?
 A: This is a nontrivial problem--the electromagnetic self-force${}^{1}$ of the charged particle will cause it to deviate from geodesic motion by an amount with involves the integral of its path.  This was worked out by Bryce DeWitt in the '60s, but I don't have a reference without hunting around for it.
Your initial equations are set up correctly, however.  You would, in principle, need another equation to determine $F_{ab}$ as a function of time, and you would need an equation telling you what the background spacetime is, so you can define $\nabla_{a}$.
${}^{1}$For those who don't know, the electromagnetic self force happens when an accelerating particle interacts with the electromagnetic field that it, itself, creates.  You can also interpret it as the "recoil" of the electromagnetic radiation that it produces by its acceleration.
A: What you wrote down is conceptually correct. However to actually make progress analytically in GR, brute force almost never works, you have to know as many tricks as possible. You can also solve the equations numerically, it shouldn't be hard in this case.
You need to specify the metric before you know what $\nabla$ is. Try writing out the christoffel symbols explicitly and you'll see that you need it.
If you really want to do physics near the surface of the earth you can use an approximately flat metric, like
\begin{equation}
ds^2=-\left(1-2\phi(r)\right) dt^2+\left(1+2\phi(r)\right)d\vec{x}^2
\end{equation}
But in this case the geodesic equation will work out to be newton's laws and you won't gain anything. You could work out $1/c$ corrections peturbatively this way. That's the most that would come up in most situations involving actual observables. However it also won't teach you much about GR per se.
So I propose you work around the Schwarzchild metric. To get the geodesics relevant for the neutral particle, the best technique here is to use the killing equation and the requirement that $g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=-1$. You can read about this in chapter 7 of http://preposterousuniverse.com/grnotes/. You might not be able to get a full, closed expression for the radial geodesics but you can at least write them as an integral.
However I recommend computing the actual geodesic equation for schwarzchild, so you can stare at what a horrendous nonlinear mess it is, before using the killing vector method, so you can appreciate how much important these computational tricks are.
I think the background electric field problem is pretty nontrivial to solve in general, even if you assume the particle accelerates slowly enough that you can ignore the self-force. I would assume the electric fieldis small enough that you can ignore the gravitational effects of the electric field, so you can still work with a schwarzchild metric instead of needing Reisner-Nordstrom (why make your life unnecessarily complicated?). But to be correct you should use R-N of course. 
The first step is to compute the electric field; fortunately for a static spherically symmetric field you should be able to use the covariant version of the standard gauss's law trick from intro e/m. The hard thing is solving for the particle motion. You can't really use the killing vector method anymore unfortunately.  If you want an analytic solution, I would do one of three things: (1) treat the charge of the particle as small and peturb your 'forced geodesic equation' around a real schwarzchild geodesic, (2) work in the "weak field regime" (which means you are not near the event horizon of a black hole) where things will reduce to newtonian gravity + e/m, (3) work near the black hole horizon but switch to Kruskal coordinates and treat $r-2GM$ as small (then you are essentially in Rindler space and things should become more tractable). Alternatively you can just write out the exact geodesic equation with your forcing term and solve it numerically. 
Learning how to solve geodesics in schwarzchild is absolutely worth investing time thinking about. Figuring out how to solve the equations when you turn on an electric field sounds like it could be a very complicated problem and might not be very insightful at the end of the day. 
