  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.
