Let G be a locally compact group, let Omega:GxG -> C be a 2-cocycle, and let (Phi,Psi) be a complementary pair of strictly increasing continuous Young functions. We continue our investigation in  of the algebraic properties of the Orlicz space L Phi(G) with respect to the twisted convolution ? coming from Omega. We show that the twisted Orlicz algebra (L Phi(G),?) posses a bounded approximate identity if and only if it is unital if and only if G is discrete. On the other hand, under suitable condition on Omega, (L Phi(G),?) becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of (L Phi(G),?), namely amenability and Connes-amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.