Discrete-time modeling and reliable control of mechanical systems with inertia, spatial coupling, time delays, and random disturbances are of fundamental importance in mechanism and machine science, particularly for safety-critical and robotic applications. This paper investigated the nonlinear dynamics and stabilization of a class of discrete-time inertial reaction–diffusion mechanical systems subject to stochastic perturbations and time-varying delays. A strong spatio-temporal discretization framework was employed to formulate the system as second-order difference equations incorporating inertial effects, diffusive coupling, delayed interactions, and stochastic excitations. To describe long-term dynamic behavior in noisy environments, a mean-square pseudo almost periodic solution concept was introduced. Sufficient conditions were established to ensure the existence and uniqueness of such solutions via fixed-point techniques. Moreover, a feedback control strategy was developed to achieve global exponential stabilization in the mean-square sense. Explicit stability criteria were derived, revealing the effects of inertia, diffusion intensity, delay bounds, stochastic disturbance levels, and control gains on system reliability and convergence performance. Numerical simulations on a multi-joint robotic manipulator validated the theoretical results and illustrated the trade-off between convergence speed and robustness under stochastic disturbances.