Following our study of the carbon-rich giants in the HR diagram and of their luminosity function (Paper III), we investigate the pulsation data of the long period variables (LPVs) included in our sample. Pulsation modes (fundamental, overtone(s)) for carbon LPVs are identified in the period-radius diagram, making use of observed bi-periodicity in a small subsample of those stars, and of comparison to models. Mean pulsation masses are then deduced from theoretical PMR-relations, with due attention paid to a possible bias while averaging. Mean (present) pulsation masses (0.6 4.0 M.) are found to increase along the group sequence HC5 to CV6, with still larger masses possibly associated with cool extreme CV7-objects with strong mass loss and thick circumstellar shells. This is consistent with the 0.8-4 M. range of initial masses found in Paper III for the majority of carbon-rich giants affected by mass loss during their evolution. The pulsation masses found for a few HC-stars (M less than or equal to 0.8 M.) are consistent with their low initial masses (M-i less than or similar to 1.1 M.); as inferred from their thick disk membership (age similar or equal to 11 Gyr?) and locus in the HR diagram. A mean pulsation mass of similar or equal to0.6 M. is found for the three population II Cepheids in the sample. A mass-luminosity diagram is proposed for the Galactic carbon giants. The data from observations is found consistent with theoretical predictions from AGB modeling, specially the third dredge-up (TDU) through thermal pulses (TP) with a carbon star formation line (CSFL) for TP-AGB stars. It appears that the CV-giants are close to the tip and end of their evolutionary tracks in the TP-AGB of the HR diagram. It is confirmed that this end shifts toward lower effective temperatures and higher luminosities, with increasing masses. It is shown that the C/O abundance ratios do correlate with effective temperatures, according to three distinct distributions (halo CH stars, thick disk HC-stars, and thin disk CV-stars). The mean stellar density decreases along the HC5-CV7 sequence, while the surface gravity remains nearly constant at about 0.5 CGS unit (log g similar or equal to -0.3; 5 x 10(-3) SI). The nature of (thin disk) CV-stars as TP-AGB objects being confirmed, the discussion is focused on (thick disk) HC-stars since the origin of these old low-mass giants remains unclear. Unpredicted extra mixing on RGB and/or E-AGB is favored. Evolution from (old, low O/H) dwarf carbon stars is also considered since observations of metal-poor stars and recent calculations point to large supersolar [C/Fe] ratios in Population III objects and contamination through rapid cycling in the interstellar medium.
