We present a fully parameterized bead-spring chain model for stained $\lambda$-phage DNA. The model accounts for the finite extensibility of the molecule through a worm-like chain potential for the spring, as well as excluded volume effects and fluctuating hydrodynamic interactions. Parameters are determined from equilibrium experimental data for 21$\mu$m stained $\lambda$-phage DNA, and are shown to quantitatively predict the non-equilibrium behavior of the molecule. The model is then used to predict the equilibrium and non-equilibrium behavior of DNA molecules of arbitrary length (up to 126$\mu$m in this work). The effects of hydrodynamic interactions are investigated for conformational and rheological properties of interest. When the bead friction coefficient is fit to the experimental relaxation time at a particular molecular weight, both equilibrium and non-equilibrium conformational properties are adequately predicted by either the free-draining or non-free-draining model at that molecular weight. However, material properties and diffusivity are severely affected by neglecting hydrodynamic interactions, which also renders the model incapable of predicting the transient and dynamic behavior of DNA at higher molecular weights. We find that using a consistently averaged hydrodynamic interaction tensor produces results (both transient and static) that are in quantitative agreement with full fluctuating hydrodynamic interactions. We show that the shear viscosity and first normal stress coefficient follow simple scaling arguments over a wide range of Weissenberg numbers. In shear flow, we find two regimes at high shear rate ($\dot{\gamma}$) that follow different scaling behavior. In the first, the viscosity and first normal stress coefficient scale roughly as $\dot{\gamma}^{-\frac{6}{11}}$ and $\dot{\gamma}^{-\frac{14}{11}}$, respectively. At higher shear rates, these become $\dot{\gamma}^{-\frac{2}{3}}$ and $\dot{\gamma}^{-\frac{4}{3}}$. These regimes can be understood based on scaling arguments for the diffusion of chain ends.