  We compare observations of HI from the Very Large Array (VLA) and the Arecibo
Observatory and observations of HCO$^+$ from the Atacama Large
Millimeter/submillimeter Array (ALMA) and the Northern Extended Millimeter
Array (NOEMA) in the diffuse ($A_V\lesssim1$) interstellar medium (ISM) to
predictions from a photodissociation region (PDR) chemical model and
multi-phase ISM simulations. Using a coarse grid of PDR models, we estimate the
density, FUV radiation field, and cosmic ray ionization rate (CRIR) for each
structure identified in HCO$^+$ and HI absorption. These structures fall into
two categories. Structures with $T_s<40~\mathrm{K}$, mostly with
$N(\mathrm{HCO^+})\lesssim10^{12}~\mathrm{cm^{-2}}$, are consistent with modest
density, FUV radiation field, and CRIR models, typical of the diffuse molecular
ISM. Structures with spin temperature $T_s>40~\mathrm{K}$, mostly with
$N(\mathrm{HCO^+})\gtrsim10^{12}~\mathrm{cm^{-2}}$, are consistent with high
density, FUV radiation field, and CRIR models, characteristic of environments
close to massive star formation. The latter are also found in directions with a
significant fraction of thermally unstable HI. In at least one case, we rule
out the PDR model parameters, suggesting that alternative mechanisms (e.g.,
non-equilibrium processes like turbulent dissipation and/or shocks) are
required to explain the observed HCO$^+$ in this direction. Similarly, while
our observations and simulations of the turbulent, multi-phase ISM agree that
HCO$^+$ formation occurs along sightlines with
$N(\mathrm{HI})\gtrsim10^{21}~\mathrm{cm^{-2}}$, the simulated data fail to
explain HCO$^+$ column densities
$\gtrsim\rm{few}\times10^{12}~\mathrm{cm^{-2}}$. Since a majority of our
sightlines with HCO$^+$ had such high column densities, this likely indicates
that non-equilibrium chemistry is important for these lines of sight.
