We describe the results of the two methods we developed to calculate the stationary nonlinear solutions in one-dimensional plasmonic slot waveguides made of a finite-thickness nonlinear dielectric core surrounded by metal regions. These two methods are described in detail in the preceding article [Walasik et al., submitted]. For symmetric waveguides, we provide the nonlinear dispersion curves obtained using the two methods and compare them. We describe the well known low-order modes and the higher-modes that were not described before. All the modes are classified into two families: modes with and without nodes. We also compare nonlinear modes with nodes with the linear modes in similar linear slot waveguides with a homogeneous core. We recover the symmetry breaking Hopf bifurcation of the first symmetric nonlinear mode toward an asymmetric mode and we show that one of the higher modes also exhibits a bifurcation. We study the behavior of the bifurcation of the fundamental mode as a function of the permittivities of a metal and a nonlinear core. We demonstrate that the bifurcation can be obtained at low power levels in structures with optimized parameters. Moreover, we provide the dispersion curves for asymmetric nonlinear slot waveguides. Finally, we give results concerning the stability of the fundamental symmetric mode and the asymmetric mode that bifurcates from it using both theoretical argument and numerical propagation simulations from two different full-vector methods. We investigate also the stability properties of the first antisymmetric mode using our two numerical propagation methods.