We investigate the structure and stability of hypercritical accretion flows around stellar-mass black holes, taking into account neutrino cooling, lepton conservation, and using for the first time a realistic equation of state in order to properly treat the dissociation of nuclei. We obtain the radial distributions of physical properties, such as density, temperature, and electron fraction, for various mass accretion rates 0.1-10M(circle dot) s(-1). We find that, depending on mass accretion rates, different physics considerably affect the structure of the disk; the most important physics are (1) the photodissociation of nuclei around r similar to 100r(g) for relatively low mass accretion rates (M similar to 0.01Y-0.1 M(circle dot) s(-1)), (2) efficient neutrino cooling around r similar to 10rgY100rg for moderately high mass accretion rates (M similar to 0.2-1.0 M(circle dot) s(-1)), and (3) neutrino trapping (r similar to 3r(g)-10r(g)) for very high mass accretion rates (M greater than or similar to 2: 0 M(circle dot) s(-1)). We also investigate the stability of hypercritical accretion flows by drawing the thermal equilibrium curves and find that efficient neutrino cooling makes the accretion flows rather stable against both thermal and viscous modes.