We investigate the out-of-equilibrium properties of a simple quantum impurity model, the interacting resonant level model. We focus on the scaling regime, where the bandwidth of the fermions in the leads is larger than all the other energies, so that the lattice and the continuum versions of the model become equivalent. Using time-dependent density matrix renormalization group simulations initialized with states having different densities in the two leads, we extend the results of Boulat, Saleur, and Schmitteckert [Phys. Rev. Lett. 101, 140601 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.140601] concerning the current-voltage ($I-V$) curves, for several values of the interaction strength U. We estimate numerically the Kondo scale $T_B$ and the exponent b(U) associated to the tunneling of the fermions from the leads to the dot. Next, we analyze the quantum entanglement properties of the steady states. We focus in particular on the entropy rate α, describing the linear growth with time of the bipartite entanglement in the system. We show that, as for the current, α/$T_B$ is described by some function of U and of the rescaled bias V/$T_B$. Finally, the spatial structure of the entropy profiles is discussed.