Hook dynamics are important in the motility of singly flagellated bacteria during flick motility. Although the hook is relatively short, during reorientation events it may undergo large deformations, leading to nonlinear behavior. Here, we explore when these nonlinear and large deformations are important for the swimming dynamics in different ranges of hook flexibilities and flagellar motor torques. For this purpose, we investigate progressively more faithful models for the hook, starting with linear springs, then models that incorporate nonlinearities due to larger hook deformations. We also employ these models both with and without hydrodynamic interactions between the flagellum and cell body to test the importance of those hydrodynamic interactions. We show that for stiff hooks, bacteria swim with a flagellum rotating on-axis in orbits and hydrodynamic interactions between the cell body and flagellum change swimming speeds by about 40%. As the hook stiffness decreases, there is a critical hook stiffness that predicts the initiation of the dynamic instability causing flicks. We compare the transition value of stiffnesses predicted by our models to experiments and show that nonlinearity and large deflections do not significantly affect critical transition values, while hydrodynamic interactions can change transition values by up to 13%. Below the transition value, we observe precession of the flagellum, in which it deflects off-axis to undergo nearly circular stable trajectories. However, only slightly below the transition stiffness, nonlinearity in hook response destabilizes precession, leading to unstable deflections of the flagellum. We conclude that while the linear hook response can qualitatively predict transition stiffnesses, nonlinear models are necessary to capture the behavior of hooks for stiffnesses below transition. Furthermore, we show that for the lower range of hook stiffnesses observed in actual bacteria, models which capture the full deformations of hooks are necessary. Inclusion of the hydrodynamic interactions of the cell body, hook, and flagellum is required to quantitatively simulate nonlinear dynamics of soft hooks during flick motility.