We examine how the uncertainties involved in supernova dynamics, as well as in nuclear data inputs, affect the nu p-process in the neutrino-driven winds. For the supernova dynamics, we find that the wind termination by the preceding dense ejecta shell, as well as the electron fraction (Y-e,Y-3; at 3 x 109 K), plays a crucial role. A wind termination within the temperature range of (1.5-3) x 10(9) K greatly enhances the efficiency of the nu p-process. This implies that the early wind phase, when the innermost layer of the preceding supernova ejecta is still similar to 200-1000 km from the center, is most relevant to the nu p-process. The outflows with Y-e,Y-3 = 0.52-0.60 result in the production of the p-nuclei up to A = 108 with interesting amounts. Furthermore, the p-nuclei up to A = 152 can be produced if Y-e,Y-3 = 0.65 is achieved. For the nuclear data inputs, we test the sensitivity to the rates relevant to the breakout from the p-p chain region (A < 12), to the (n, p) rates on heavy nuclei, and to the nuclear masses along the nu p-process pathway. We find that a small variation of the rates of triple-a and of the (n, p) reaction on Ni-56 leads to a substantial change in the p-nuclei production. We also find that Pd-96 (N = 50) on the nu p-process path plays a role as a second seed nucleus for the production of heavier p-nuclei. The uncertainty in the nuclear mass of Zr-82 can lead to a factor of two reduction in the abundance of the p-isotope Sr-84.