The electronic and magnetic properties of the new hydride superconductor CaFeAsH, which superconducts up to 47 K when electron-doped with La, and the isovalent alloy system CaFeAsH$_{1-x}$F$_x$ are investigated using density functional based methods. The $vec Q = (pi,pi,0)$ peak of the nesting function $xi(vec q)$ is found to be extremely strong and sharp, and additional structure in $xi(vec q)$ associated with the near-circular Fermi surfaces (FSs) that may impact low energy excitations is quantified. The unusual band introduced by H, which shows strong dispersion perpendicular to the FeAs layers, is shown to be connected to a peculiar van Hove singularity just below the Fermi level. This band provides a three dimensional electron ellipsoid Fermi surface not present in other Fe-based superconducting materials nor in CaFeAsF. Electron doping by 25% La or Co has minor affect on this ellipsoid Fermi surface, but suppresses FS nesting strongly, consistent with the viewpoint that eliminating strong nesting and the associated magnetic order allows high T$_c$ superconductivity to emerge. Various aspects of the isovalent alloy system CaFeAsH$_{1-x}$F$_x$ and means of electron doping are discussed in terms of influence of incipient bands.
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