Sorry for my late reply.
Yes, its fully vectorial 2D solver in frequency domain that uses Yee cells to calculate electromagnetic fields. You can read more about this here
Not exactly. In EME, eigenmodes are calculated in the cross section of each cells (similar to as FDE) and then propagated within each cell. Then scattering matrices are calculated on each boundary. This technique is fully vectorial and considers both forward and backward propagating modes. You can read more about this here.
As you realized, since EME relies on 2D FDE solver it is not a 3D eigenmode solver. This solver is quite strong if you want to propagate modes inside a long waveguide for example.
If you can provide me with the specific application of your interest, we can discuss it further and decide on what software to use.
FDTD might be a good approach for some of the 3D Eigenmode solver applications.