The da Vinci Research Kit (dVRK) is widely used for research in robot-assisted surgery, but most modeling and control methods target the first-generation dVRK Classic. The recently introduced dVRK-Si, built from da Vinci Si hardware, features a redesigned Patient Side Manipulator (PSM) with substantially larger gravity loading, which can degrade control if unmodeled. This paper presents the first complete kinematic and dynamic modeling framework for the dVRK-Si PSM. We derive a modified DH kinematic model that captures the closed-chain parallelogram mechanism, formulate dynamics via the Euler-Lagrange method, and express inverse dynamics in a linear-in-parameters regressor form. Dynamic parameters are identified from data collected on a periodic excitation trajectory optimized for numerical conditioning and estimated by convex optimization with physical feasibility constraints. Using the identified model, we implement real-time gravity compensation and computed-torque feedforward in the dVRK control stack. Experiments on a physical dVRK-Si show that the gravity compensation reduces steady-state joint errors by 68-84% and decreases end-effector tip drift during static holds from 4.2 mm to 0.7 mm. Computed-torque feedforward further improves transient and position tracking accuracy. For sinusoidal trajectory tracking, computed-torque feedforward reduces position errors by 35% versus gravity-only feedforward and by 40% versus PID-only. The proposed pipeline supports reliable control, high-fidelity simulation, and learning-based automation on the dVRK-Si.