CO(7-6) and [C I](2-1) survey in z > 6 quasars

High-redshift ($z\gtrsim6$) quasars trace the earliest supermassive black holes and intense star formation, offering key laboratories for black hole-galaxy evolution at cosmic dawn. While far-infrared studies have revealed large dust reservoirs and strong [C II] emission, the physical conditions and molecular gas content of their ISM remain uncertain. We present ALMA Band 3 observations of the redshifted CO(7-6) and [C I](2-1) emission lines and dust continuum in a sample of 18 quasars at $z \sim 6$. We detected CO(7-6) in 15/18, [C I](2-1) in 6/18, and continuum in 13/18 sources. Line luminosities and continuum fluxes were used to estimate molecular gas masses from CO, [C I], and dust, and a hierarchical Bayesian cross-calibration of all four tracers yielded consistent per-source $M_{\rm H_2}$ estimates and conversion factors. Comparison with PDR and XDR model grids using the $L_{\rm [CII]}/L_{\rm [CI]}$ and $L_{\rm CO(7--6)}/L_{\rm TIR}$ ratios suggests gas densities of $n > 10^4$ cm$^{-3}$ and radiation fields of $G_0 \sim 10^3$--$10^4$ for sources consistent with PDR solutions, while many quasars fall outside the model parameter space. The $L'_{\rm CO(7-6)}/L'_{\rm [CI](2-1)}$ ratio indicates that a large fraction of the molecular gas resides in a warm and highly excited phase. Together these results suggest that classical PDR heating alone cannot explain the observed line ratios and that additional volumetric processes such as X-ray irradiation, turbulence and shocks, or enhanced cosmic-ray heating likely influence the excitation of the cold ISM. They demonstrate the power of multi-line diagnostics in revealing the excitation and structure of the cold ISM in early quasar host galaxies and highlight the need for joint analysis of CO, [C I], [C II], and dust emission to characterize star formation and AGN-driven heating at cosmic dawn.