In this paper, we establish connections between the thresholds and mechanisms of the damage and white-light generation upon femtosecond laser irradiation of wide-bandgap transparent materials. On the example of Corning Willow glass, evolution of ablation craters, their quality, and white-light emission were studied experimentally for 130-fs, 800-nm laser pulses. The experimental results indicate co-existence of several ablation mechanisms which can be separated in time. Suppression of the phase explosion mechanism of ablation was revealed at the middle of the irradiation spots. At high laser fluences, air ionization was found to strongly influence ablation rate and quality and the main mechanisms of the influence are analysed. To gain insight into the processes triggered by laser radiation in glass, numerical simulations have been performed with accounting for the balance of laser energy absorption and its distribution/redistribution in the sample, including bremsstrahlung emission from excited free-electron plasma. The simulations have shown an insignificant role of avalanche ionization at such short durations of laser pulses while pointing to high average energy of electrons up to several dozens of eV. At multi-pulse ablation regimes, improvement of crater quality was found as compared to single/few pulses.