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 km s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, (2) the X-ray heating efficiency of early sources to $fX>0.0042$, and (3) the low-energy cutoff of the X-ray spectral energy distribution to keV. We also constrain the star-formation efficiency ($f\ast$), the electron scattering optical depth ($\tau e$), and the mean-free path of ionizing photons ($Rmfp$). 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 $\tau e$ 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 and $Rmfp>27.5$ Mpc. The external observations also have a significant effect on $Vc$ due to its degeneracy with $\tau e$, while the constraints on $f\ast$, $fX$, and $\nu min$, remain primarily determined by EDGES.