> 1) [...] Is it possible / likely for solve to use currents from neighboring cells?
No, most probably not. Looking at Maxwell's equations, dE/dt = -j + curl B, so updating E requires j at the same position, but B in a neighborhood. The other Maxwell equation is dB/dt = - curl E, so updating B requires E in a neighborhood.
> 2) Considering the amount of data necessary to transfer: GATHER (and also SOLVE) only uses the Ex and Ey from the grid, so I suppose it is sufficient to share just these and not all the grid properties?
If you mean, Ex, Ey, and Bz, then yes, you only need those.