This article, which is a continuation of a previous paper, is about the study of viscosity solutions for second order fully nonlinear parabolic equations, having a L¹ dependence in time, associated with nonlinear Neumann boundary conditions. First, we obtain the existence of continuous viscosity solutions by adapting Perron's method and using the comparison results obtained previously. Then, we apply these existence and comparison results to the study of the level-set approach for Front propagations problems when the normal velocity has a L¹ dependence in time.