The transition density $\rho _{t} $ and pressure $P_{t} $ at the inner edge separating the liquid core from the solid crust of neutron stars are systematically studied using a modified Gogny (MDI) and $51$ popular Skyrme interactions within well established dynamical and thermodynamical methods.First of all, it is shown that the widely used parabolic approximation to the full Equation of State (EOS) of isospin asymmetric nuclear matter may lead to huge errors in estimating the transition density and pressure, especially for stiffer symmetry energy functionals $E_{sym} (\rho)$, compared to calculations using the full EOS within both the dynamical and thermodynamical methods mainly because of the energy curvatures involved.Thus, fine details of the EOS of asymmetric nuclear matter are important for locating accurately the inner edge of the neutron star crust.Secondly, the transition density and pressure decrease roughly linearly with the increasing slope parameter $L$ of the $E_{sym}(\rho)$ at normal nuclear matter density using the full EOS within both the dynamical and thermodynamical methods.It is also shown that the thickness, fractional mass and moment of inertia of neutron star crust are all very sensitive to the parameter $L$ through the transition density $\rho _{t} $ whether one uses the full EOS or its parabolic approximation.Moreover, it is shown that the $E_ {sym } (\rho)$ constrained in the same sub saturation density range as the neutron star crust by the isospin diffusion data in heavy-ion collisions at intermediate energies limits the transition density and pressure to $0.040$ fm $^{-3}$ $\leq \rho _{t}\leq 0.065$ fm$^{-3}$ and $0.01$ MeV/fm$^{3}$ $\leq P_{t}\leq 0.26$ MeV/fm$^ {3 } $, respectively.These constrained values for the transition density and pressure are significantly lower than their fiducial values currently used in the literature.Furthermore, the mass-radius relation and several other properties closely related to the neutron star crust are studied by using the MDI interaction.It is found that the newly constrained $\rho_t$ and $P_t$ together with the earlier estimate of $\Delta Ⅰ/(I)>0.014$ for the crustal fraction of the moment of inertia of the Vela pulsar impose a more stringent constraint of SR\geq 4.7+4.0M/M_{\odot} $ km for the radius $R$ and mass $M$ of neutron stars compared to previous studies in the literature.