We introduce and analyze a quasilinear size-structured population model with nonlinearities accounting for nonlocal interactions between individuals. The recruitment (immigration), growth and death rates are inhomogeneous in time and/or space and depend on weighted averages of the density. We first prove the existence and uniqueness of globally bounded weak solutions using the characteristic curves and Banach fixed point Theorem, after transforming the partial differential equation into an equivalent system of integral equations. We then investigate the long-time behavior of the PDE in the case when the growth rate is separable. Applying a classical time-scaling transformation, the problem boils down to a PDE with linear growth rate and nonlinear inflow boundary condition, entering the theoretical framework of abstract semilinear Cauchy problems. We can then perform a bifurcation analysis which reveals the richness of the model behavior. Depending on the ratio of the recruitment to the growth rate, the model can exhibit multistability and stable oscillatory solutions, emanating respectively through saddle-node and Hopf bifurcations. We illustrate these theoretical results on the biological application motivating this work, oogenesis, the process of production and maturation of female gametes (oocytes) that is critical to reproductive fitness.