We present a novel, self-consistent, semi-numeric Cosmic Dawn (CD) simulation in which small-scale star formation (SF) is calibrated to the \emph{AEOS} and \emph{Renaissance} hydrodynamic simulations. SF proceeds within dark matter (DM) halos via neural network emulation while considering large-scale fluctuations in density and feedback. We translate the resulting 3D distribution of galaxies into predictions for the 21-cm brightness temperature, \Tb, and power spectrum, \PS. We simulate several unique realizations to study the impact of varying astrophysics on \Tb, finding that more efficient Population II (PopII) SF largely yields stronger Lyman-$\alpha$ coupling, resulting in a shallower and wider absorption trough. However, we find that PopII SF dominates \PS\ at $z \lesssim 20$ and on smaller scales at intermediate redshifts ($k \gtrsim 0.2\ \mathrm{Mpc^{-1}}$ at $z \simeq 34-20$) while Population III (PopIII) SF dominates \PS\ at $z\gtrsim34$ and on larger scales at intermediate redshifts. Compared with previous works, we find that the combination of hydrodynamic SF calibration, a critical halo mass for SF considering \Htwo\ self-shielding, and stochastic DM halo merger histories results in both earlier SF and higher SF rates across CD. Further, we find that the delay period separating PopIII and PopII SF (\tdelay) significantly impacts \Tb, and that one must include DM halo merger histories to properly account for this transition. Finally, we find our fiducial \Tb\ to be detectable at $z\lesssim25$ with 1080 hours of HERA observations under moderate foreground assumptions, and the lack of such a detection at $z \gtrsim 20$ would suggest \tdelay\ $\gtrsim$ 30 Myr.