The glib wrong answer to your first question is that Reinich Misner Wheeler theory is *source free*, so there is no equivalence principle for the motion of test charges, as there are no charges. This glib answer is wrong, because there are still electrovac black holes in the theory which serve as charges.

For the second question, What accounts for electromagnetism is that the vacuum equations aren't satisfied $R_{\mu\nu}\ne 0$, rather there is a nonzero stress tensor, which is the Einstein tensor found from the given solution. This stress tensor is equal to that produced by some F field obeying the free-space Maxwell equation.

Coming back to question 1--- not every geometric field can be reduced to 0 by a local coordinates, just the connection coefficients. Only the effect of the connection coefficients can be zeroed out. The curvature is covariant, and if it is nonzero in one frame, it is nonzero in all frames. A solution to the RMW equation gives you a metric compatible with a unique F tensor, determined up to coordinate covariance by the metric tensor.

How do we know that the F tensor is unique? Here is a physical argument, not a mathematical proof. You can always deform the given solution by adding the metric for a local charged black hole to the given solution of the theory. The result will have a deflecting black hole, and by studying the deflections of the infinitesimally perturbed solutions with a test-charge black-hole, you can uniquely determine the F tensor of the given solution, by looking at infinitesimally different solution. This argument can surely be turned into a proof, it would require the appropriate differentiation of the metric in the direction corresponding to adding an infinitesimal black hole solution, but for physics purposes this is sufficient to see why the answer must be unique (**Edit**: unique up to E-B duality rotation, with the corresponding electric-magnetic charge rotation for the test black hole--- this is obvious and mentioned in the other answer, but I didn't notice).