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 $ u_{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 $ u_{rm min}$, remain primarily determined by EDGES.
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