We theoretically investigated the stability of highly charged C(60) (z+) cations produced from C(60) with an ultrashort intense laser pulse of lambda approximately 1800 nm. We first calculated the equilibrium structures and vibrational frequencies of C(60) (z+) as well as C(60). We then calculated key energies relevant to dissociation of C(60) (z+), such as the excess vibrational energy acquired upon sudden tunnel ionization from C(60). By comparing the magnitudes of the calculated energies, we found that C(60) (z+) cations up to z approximately 12 can be produced as a stable or quasistable (microsecond-order lifetime) intact parent cation, in agreement with the recent experimental report by V. R. Bhardwaj et al. [Phys. Rev. Lett. 93, 043001 (2004)] that almost only intact parent C(60) (z+) cations up to z=12 are detected by a mass spectrometer. The results of Rice-Ramsperger-Kassel-Marcus calculation suggest that the lifetime of C(60) (z+) drastically decreases by ten orders of magnitude as z increases from z=11 to z=13. Using the time-dependent adiabatic state approach, we also investigated the vibrational excitation of C(60) and C(60) (z+) by an intense near-infrared pulse. The results indicate that large-amplitude vibration with energy of >10 eV is induced in the delocalized h(g)(1)-like mode of C(60) (z+).