Addressing the frequent occurrence of landing gear shimmy under maneuvering conditions， this paper conducts research on high-fidelity modeling and parameter coupling effects. Based on a delay tire dynamic model， the coupled landing gear—tire shimmy dynamics equations are developed， which incorporate the time-varying tire lateral deformation modes. Utilizing Hopf bifurcation theory， we compute the global dynamic characteristics of the system through multi-dimensional projections， quantitatively revealing the influence patterns and mechanisms of key parameters （such as lateral bending stiffness and torsional stiffness） on the stability of the landing gear system across various taxiing speeds. Comparative results with traditional tire models show consistency of both and limitations of traditional tire models in capturing parameter coupling effects. By employing an enhanced Bayesian optimization method， we optimize the bifurcation characteristics of the landing gear system， resulting in a significant improvement in stability. Based on the parameter optimization results and combined with Sobol global sensitivity analysis， the comparative effects of multi-dimensional parameters on system stability are quantitatively revealed， with the results complementing and validating those from the Hopf bifurcation analysis. The study identifies that when the torsional mode frequency of the landing gear approaches the lateral bending mode frequency， substantial modal coupling occurs， creating a bistable shimmy critical boundary that drastically reduces stability. Low lateral stiffness of the landing gear may trigger shimmy during low-speed taxiing， while greater lateral stiffness at high speeds necessitates matched damping enhancement for shimmy mitigation. Increasing torsional stiffness and decreasing tire lateral stiffness yield notable shimmy suppression effects across the entire speed spectrum， indicating their potential as key control parameters for targeted design and optimization.