To overcome the limitation of conventional nose landing gear shimmy damper design, which is mainly based on the critical damping for torsional shimmy while neglecting the high-damping-side instability boundary, this study establishes a nonlinear shimmy dynamic model of a nose landing gear equipped with a hydraulic damper and an inerter-based suppression system. Hopf bifurcation analysis, numerical continuation, and energy-evolution analysis are employed to investigate the effects of damper, structural, and inerter parameters on the zero-shimmy region over the full speed range. The results show that the zero-shimmy damping interval is not governed by a single torsional critical damping value but is jointly bounded by the tire-induced torsional shimmy boundary on the low-damping side and the lateral or structural torsional shimmy boundary on the high-damping side. Thus, increasing damper damping is not always beneficial. Among the structural parameters, trail determines the existence of the zero-shimmy region, strut torsional stiffness mainly regulates the high-damping-side boundary, and rake angle provides local correction. With proper inertance and tuning-stiffness matching, the inerter-based system raises the upper critical damping by about 210% and improves low-speed shimmy suppression by transferring vibration energy to the damping branch through inertial coupling.