We present an experimental and numerical study of the damage and ablation thresholds at the surface of a dielectric material, e.g., fused silica, using short pulses ranging from 7 to 300 fs. The relevant numerical criteria of damage and ablation thresholds are proposed consistently with experimental observations of the laser irradiated zone. These criteria are based on lattice thermal melting and electronic cohesion temperature, respectively. The importance of the three major absorption channels (multi-photon absorption, tunnel effect, and impact ionization) is investigated as a function of pulse duration (7-300 fs). Although the relative importance of the impact ionization process increases with the pulse duration, our results show that it plays a role even at short pulse duration (<50 fs). For few optical cycle pulses (7 fs), it is also shown that both damage and ablation fluence thresholds tend to coincide due to the sharp increase of the free electron density. This electron-driven ablation regime is of primary interest for thermal-free laser-matter interaction and therefore for the development of high quality micromachining processes.