  Insulating FeGa$_3$ poses peculiar puzzles beyond the occurrence of an
electronic gap in an intermetallic compound. This Fe-based material has a very
distinctive structural characteristic with the Fe atoms occurring in dimers.
The insulating gap can be described comparably well in either the weakly
correlated limit or the strongly correlated limit within density functional
theory viewpoints, where the latter corresponds to singlet formation on the
Fe$_2$ dimers. Though most of the calculated occupied Wannier functions are an
admixture of Fe $3d$ and Ga $4s$ or $4p$ states, there is a single bonding-type
Wannier function per spin centered on each Fe$_2$ dimer. Density functional
theory methods have been applied to follow the evolution of the magnetic
properties and electronic spectrum with doping, where unusual behavior is
observed experimentally. Both electron and hole doping are considered, by Ge
and Zn on the Ga site, and by Co and Mn on the Fe site, the latter introducing
direct disturbance of the Fe$_2$ dimer. Results from weakly and strongly
correlated pictures are compared. Regardless of the method, magnetism including
itinerant phases appears readily with doping. The correlated picture suggests
that in the low doping limit Mn (for Fe) produces an in-gap hole state, while
Co (for Fe) introduces a localized electronic gap state.
