Families of 3D stable solitons and vortices as well as robust dynamical modes of precession are found in a 3D media with the local strength of the cubic self-repulsion growing from the center to periphery. In addition (3D) stable vortex dipoles of uni-charge as well as completely novel stable dipoles with antipodal topological charges together with hybrid dipoles consisting of 3D solitons and 3D vortices sharing the same vertical axis will be presented in a model with peanut-shaped strong nonlinear traps with uniform and strongly non-uniform nonlinearities.