We study the physical conditions in damped Lyman-$\alpha$ systems (DLAs), using a sample of 33 systems toward 26 QSOs acquired for a recently completed survey of H$_2$ by Ledoux et al. (2003). H$_2$ is detected in 13-20% of the DLAs in our sample. Using the rotation level populations of H$_2$ and fine-structure excitations of C I we show the mean kinetic temperature of H$_2$ components is 153$\pm$78 K, $n_{\rm H}$ = 10$-$250 cm$^{-3}$, and the ambient radiation field is similar to or slightly higher than the mean diffuse UV field of the Galaxy. Combining this with the success rate of detecting H$_2$ in DLAs we conclude that at least 13$-$20% of DLAs at $z_{\rm abs}\ge1.9$ show the presence of CNM and substantial star-formation activity. C II$^*$ absorption is detected in all the components where H$_2$ absorption is seen. The level populations of C II in these systems is consistent with the physical parameters derived from the excitation of H$_2$ and C I. We detect C II$^*$ in about 50% of the DLAs and therefore in a considerable fraction of DLAs that do not show H$_2$. The absence of C I absorption, the measured N(C II$^*$)/N(C II) and N(Al III)/N(Al II) ratios in these systems are consistent with the gas having lower density (n$_{\rm H} = 0.3-6 cm^{-3}$) than that seen in the H$_2$ components. 50% of the DLAs that do not show C II$^*$ are consistent with them originating from a low density warm neutral medium.