Optimal estimation is a promising tool for estimation of payloads' inertial parameters and localization of robots in the presence of multiple contacts. To harness its advantages in robotics, it is crucial to solve these large and challenging optimization problems efficiently. To tackle this, we (i) develop a multiple shooting solver that exploits both temporal and parametric structures through a parametrized Riccati recursion. Additionally, we (ii) propose an inertial manifold that ensures the full physical consistency of inertial parameters and enhances convergence. To handle its manifold singularities, we (iii) introduce a nullspace approach in our optimal estimation solver. Finally, we (iv) develop the analytical derivatives of contact dynamics for both inertial parametrizations. Our framework can successfully solve estimation problems for complex maneuvers such as brachiation in humanoids, achieving higher accuracy than conventional least squares approaches. We demonstrate its numerical capabilities across various robotics tasks and its benefits in experimental trials with the Go1 robot.