  Observations of the 21-cm signal are opening a window to the cosmic-dawn
epoch, when the first stars formed. These observations are usually interpreted
with semi-numerical or hydrodynamical simulations, which are often
computationally intensive and inflexible to changes in cosmological or
astrophysical effects. Here, we present an effective, fully analytic model for
the impact of the first stars on the 21-cm signal, using the modular code
Zeus21. Zeus21 employs an analytic prescription of the star formation rate
density (SFRD) to recover the fully nonlinear and nonlocal correlations of
radiative fields that determine the 21-cm signal. We introduce the earliest
Population III (Pop III) stars residing in low-mass molecular-cooling galaxies
in Zeus21, with distinct spectra from later Pop II stars. We also
self-consistently model feedback in the form of $H_2$-dissociating Lyman-Werner
(LW) radiation, as well as dark matter-baryon relative velocities, both of
which suppress star formation in the lowest-mass halos. LW feedback produces a
scale-dependence on the SFRD fluctuations, due to the long mean free path of LW
photons. Relative velocities give rise to "wiggles" in the spatial distribution
of the 21-cm signal; we present an improved calculation of the shape of these
velocity-induced acoustic oscillations, showing they remain a standard ruler at
cosmic dawn. Our improved version of Zeus21 predicts the 21-cm global signal
and power spectra in agreement with simulations at the $\sim 10\%$ level, yet
is at least three orders of magnitude faster. This public code represents a
step towards efficient and flexible parameter inference at cosmic dawn,
allowing us to predict the first billion years of the universe in mere seconds.
