The fist part of your question has a negative answer, since both Fano plane (7₃) and Moebius-Kantor configuration (8₃) are cyclic configurations. Here it is shown that the cyclic covering graphs over a dipole with girth at least 6 are exactly Levi graphs of combinatorial cyclic configurations. In your terminology, each Levi graph of a cyclic configuration has a "voltage-graph representation". Note: a dipole is a graph consisting of two vertices and a number of parallel edges between them.

If v = uw is a composite number, it is sometimes possible to find a suitable voltage graph on 2u vertices and voltages from the cyclic group Z_w to produce a straight-line drawing of the corresponding polycyclic configuration (v₃). Unfortunately this method does not apply to cyclic configurations (v_k), for v prime and is not certainly not understood well for k > 4.

The fist part of your question has a negative answer, since both Fano plane (7₃) and Moebius-Kantor configuration (8₃) are cyclic configurations. Here it is shown that the cyclic covering graphs over a dipole with girth at least 6 are exactly Levi graphs of combinatorial cyclic configurations. In your terminology, each Levi graph of a cyclic configuration has a "voltage-graph representation". Note: a dipole is a graph consisting of two vertices and a number of parallel edges between them. In particular both the Heawood graph and the Moebius-Kantor graph are counterexaples.

If v = uw is a composite number, it is sometimes possible to find a suitable voltage graph on 2u vertices and voltages from the cyclic group Z_w to produce a straight-line drawing of the corresponding polycyclic configuration (v₃). Unfortunately this method does not apply to cyclic configurations (v_k), for v prime and is not certainly not understood well for k > 4. I am not sure what happens if pseudolines are admitted in such drawings having rotational symmetry.