We prove that if A is a σ-unital exact C∗-algebra of real rank zero, then every state on K0(A) extends to a 2-quasitrace on A. This yields a generalisation of Rainone’s work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for a twisted C∗-algebra of a cofinal k-graph with no sources, stable finiteness is independent of the twisting cocycle. We also study pure infiniteness of twisted higher rank graph C∗-algebras.