They are almost Maxwell, but not quite - they are invariant under hyperbolic rotations and translations, but not under boosts, because two of the quantities in the equations don't transform in the necessary way.
Actually no, they can't. The GEM field is a rank-3 tensor, which arises from a rank-2 potential, giving the Lagrangian density
\[\mathcal{L} =-\frac{1}{16\pi } F_{\mu \nu \alpha } F^{\mu \nu \alpha } -\frac{G}{c} J^{\nu \alpha } A_{\nu \alpha }\]
This is different than would be the case for electromagnetism (which arises from a vector potential, and has a rank-2 field tensor). The above is a good approximation for the linear part of the Einstein equations, so it is a weak field approximation to GR.
Hm, I don't think this is obvious at all. For one thing, since it is not Lorentz invariant, it's also not CPT invariant, so adding this into the Standard Model is not trivial. Renormalizability also cannot just be assumed, this will have to be specifically checked; I don't know if anyone has done these (pretty complex) maths for the case of GEM.