Natural gas hydrates (NGHs) represent a strategic clean energy resource; however, their commercial exploitation is primarily constrained by significant geological risks and low recovery efficiency. In this study, a comprehensive thermal-hydrological-mechanical-chemical (THMC) coupled model, integrated with the Drucker-Prager elastoplastic constitutive criterion, was developed to prioritize the investigation of geomechanical stability under horizontal-well depressurization. The influence of bottom-hole pressure (BHP), vertical positioning of the Producer Horizontal Section (PHS) and underburden thermal convection on formation integrity was systematically evaluated to balance production efficiency and geological risks. While a four-stage gas production evolution was observed, the results emphasize that geomechanical hazards are the primary limiting factor for aggressive drawdown strategies. Although lowering BHP enhances recovery, it exacerbates sediment compaction and threatens seabed stability. Conversely, maintaining BHP above a critical threshold of 4–5 MPa was identified as essential for preserving reservoir integrity. Furthermore, PHS placement was determined to be a critical control for strain localization; a mid-reservoir location was found to optimize the trade-off between decomposition-induced weakening and heat transferred from underlying layers. Geomechanical analysis further revealed that subsidence is most pronounced near the wellbore, with high-risk deformation occurring at the reservoir–overburden interface. These findings underscore the necessity of integrating real-time deformation monitoring with adaptive well designs to mitigate geohazards during the exploitation of deep-sea NGH resources.