In the cold neutral medium, high out-of-equilibrium temperatures are created by intermittent dissipation processes, including shocks, viscous heating, and ambipolar diffusion. The high-temperature excursions are thought to explain the enhanced abundance of CH$^{+}$ observed along diffuse molecular sight-lines. Intermittent high temperatures should also have an impact on H$_2$ line luminosities. We carry out simulations of MHD turbulence in molecular clouds including heating and cooling, and post-process them to study H$_2$ line emission and hot-gas chemistry, particularly the formation of CH$^+$. We explore multiple magnetic field strengths and equations of state. We use a new H$_2$ cooling function for $n_{rm H} leq 10^5,{rm cm}^{-3}$, $Tleq 5000,{rm K}$, and variable H$_2$ fraction. We make two important simplifying assumptions: (i) the ${rm H}_2/{rm H}$ fraction is fixed everywhere, and (ii) we exclude from our analysis regions where the ion-neutral drift velocity is calculated to be greater than 5 km/s. Our models produce H$_2$ emission lines in accord with many observations, although extra excitation mechanisms are required in some clouds. For realistic r.m.s. magnetic field strengths ($approx 10$ $mu$G) and velocity dispersions, we reproduce observed CH$^+$ abundances. These findings contrast with those of Valdivia et al. (2017). Comparison of predicted dust polarization with observations by {it Planck} suggests that the mean field $gtrsim 5 mu$G, so that the turbulence is sub-Alfvenic. We recommend future work treating ions and neutrals as separate fluids to more accurately capture the effects of ambipolar diffusion on CH$^+$ abundance.
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