The tracial cyclic Rokhlin property for automorphisms of simple not necessarily unital C^*-algebras is investigated. We show that the tracial cyclic Rokhlin property is preserved by going to certain restrictions to subalgebras and taking direct limit or tensor products of actions. We also show that under certain conditions properties such as real rank zero, the tracial rank zero, stable rank one, (tracial) Z-stability, Property (SP), strict comparison on projections are passed to crossed products under automorphisms with the tracial cyclic Rokhlin property.