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-α radiation: Ly-α heating, CMB heating (mediated by Ly-α photons), and multiple scattering of Ly-α 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-α and CMB heating raise the gas temperature by up to 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 ≈ 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-α 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-α photons is important, leading to an amplification of the power spectrum by a factor of ∼ 2 − 5.