  We present the evolution of the 21-cm signal from cosmic dawn and the epoch
of reionization (EoR) in an upgraded model including three subtle effects of
Ly-a radiation: Ly-a heating, CMB heating (mediated by Ly-a photons), and
multiple scattering of Ly-a photons. Taking these effects into account we
explore a wide range of astrophysical models and quantify the impact of these
processes on the global 21-cm signal and its power spectrum at observable
scales and redshifts. We find that, in agreement with the literature, Ly-a and
CMB heating raise the gas temperature by up to $\mathcal{O}(100)$ degrees in
models with weak X-ray heating and, thus, suppress the predicted 21-cm signals.
Varying the astrophysical parameters over broad ranges, we find that in the
upgraded model the absorption trough of the global signal reaches a lowest
floor of $-165$ mK at redshifts $z\approx 15-19$. This is in contrast with the
predictions for a pure adiabatically cooling Universe, for which the deepest
possible absorption is a monotonically decreasing function of cosmic time and
is $-178$ mK at $z = 19$ and $-216$ mK at $z=15$, dropping to even lower values
at lower redshifts (e.g. $-264$ mK at $z = 10$). With the Ly-a and CMB heating
included we also observe a strong suppression in the low-redshift power
spectra, with the maximum possible power (evaluated over the ensemble of
models) attenuated by a factor of $6.6$ at $z=9$ and $k = 0.1$ Mpc$^{-1}$.
Finally, we find that at high redshifts corresponding to cosmic dawn, the
heating terms have a subdominant effect while multiple scattering of Ly-a
photons is important, leading to an amplification of the power spectrum by a
factor of $\sim 2-5$.
