  We present new constraints on parameters of cosmic dawn and the epoch of
reionization derived from the EDGES High-Band spectrum ($90-190$ MHz). The
parameters are probed by evaluating global $21$ cm signals generated with the
recently developed Global21cm tool. This tool uses neural networks trained and
tested on $\sim 30,000$ spectra produced with semi-numerical simulations that
assume the standard thermal evolution of the cosmic microwave background and
the intergalactic medium. From our analysis, we constrain at $68\%$ (1) the
minimum virial circular velocity of star-forming halos to $V_{\rm c}<19.3$ km
s$^{-1}$, (2) the X-ray heating efficiency of early sources to $f_{\rm
X}>0.0042$, and (3) the low-energy cutoff of the X-ray spectral energy
distribution to $\nu_{\rm min}<2.3$ keV. We also constrain the star-formation
efficiency ($f_*$), the electron scattering optical depth ($\tau_{\rm e}$), and
the mean-free path of ionizing photons ($R_{\rm mfp}$). We re-compute the
constraints after incorporating into the analysis four estimates for the
neutral hydrogen fraction from high-$z$ quasars and galaxies, and a prior on
$\tau_{\rm e}$ from Planck $2018$. The largest impact of the external
observations is on the parameters that most directly characterize reionization.
Specifically, we derive the combined $68\%$ constraints $\tau_{\rm e}<0.063$
and $R_{\rm mfp}>27.5$ Mpc. The external observations also have a significant
effect on $V_{\rm c}$ due to its degeneracy with $\tau_{\rm e}$, while the
constraints on $f_*$, $f_{\rm X}$, and $\nu_{\rm min}$, remain primarily
determined by EDGES.
