Context. This is the second paper of a series making use of Herschel/PACS spectroscopy of evolved stars in the THROES catalogue to study the inner warm regions of their circumstellar envelopes (CSEs). Aims. We analyse the CO emission spectra, including a large number of high-J CO lines (from J = 14-13 to J = 45-44, nu = 0), as a proxy for the warm molecular gas in the CSEs of a sample of bright carbon-rich stars spanning different evolutionary stages from the asymptotic giant branch to the young planetary nebulae phase. Methods. We used the rotational diagram (RD) technique to derive rotational temperatures (T-rot) and masses (M-H2) of the envelope layers where the CO transitions observed with PACS arise. Additionally, we obtained a first order estimate of the mass-loss rates and assessed the impact of the opacity correction for a range of envelope characteristic radii. We used multi-epoch spectra for the well-studied C-rich envelope IRC+10216 to investigate the impact of CO flux variability on the values of T-rot and M-H2. Results. The sensitivity of PACS allowed for the study of higher rotational numbers than before indicating the presence of a significant amount of warmer gas (similar to 200-900 K) that is not traceable with lower J CO observations at submillimetre/millimetre wavelengths. The masses are in the range M-H2 similar to 10(-2)-10(-5)M(Theta), anticorrelated with temperature. For some strong CO emitters we infer a double temperature (warm T-rot similar to 400 K and hot T-rot similar to 820 K) component. From the analysis of IRC+10216, we corroborate that the effect of line variability is perceptible on the T-rot of the hot component only, and certainly insignificant on M-H2 and, hence, the mass-loss rate. The agreement between our mass-loss rates and the literature across the sample is good. Therefore, the parameters derived from the RD are robust even when strong line flux variability occurs, and the major source of uncertainty in the estimate of the mass-loss rate is the size of the CO-emitting volume.