A consistent Hartree-Fock study of the equation of state (EOS) of asymmetric nuclear matter at finite temperature has been performed using realistic choices of the effective, density dependent nucleon-nucleon (NN) interaction, which were successfully used in different nuclear structure and reaction studies. Given the importance of the nuclear symmetry energy in the neutron star formation, EOS's associated with different behaviors of the symmetry energy were used to study hot asymmetric nuclear matter. The slope of the symmetry energy and nucleon effective mass with increasing baryon density was found to affect the thermal properties of nuclear matter significantly. Different density dependent NN interactions were further used to study the EOS of hot protoneutron star (PNS) matter of the $npe\mu\nu$ composition in $\beta$-equilibrium. The hydrostatic configurations of PNS in terms of the maximal gravitational mass $M_{\rm max}$ and radius, central density, pressure and temperature at the total entropy per baryon $S/A= 1,2$ and 4 have been determined in both the neutrino-free and neutrino-trapped scenarios. The obtained results show consistently a strong impact of the symmetry energy and nucleon effective mass on thermal properties and composition of hot PNS matter. $M_{\rm max}$ values obtained for the (neutrino-free) $\beta$-stable PNS at $S/A=4$ were used to assess time $t_{\rm BH}$ of the collapse of 40 $M_\odot$ protoneutron progenitor to black hole, based on a correlation between $t_{\rm BH}$ and $M_{\rm max}$ found from the hydrodynamic simulation by Hempel {\it et al.}.