In this paper, we propose a framework for energy-efficient (EE) design in reconfigurable intelligent surface (RIS)-assisted multi-satellite Internet of Things (IoT) networks, taking into account the imperfect channel state information (CSI). In this framework, the multi-satellite network is used to enhance communication capabilities, while the RIS is deployed to further improve EE performance. Our objective is to maximize the EE of the proposed network by jointly optimizing the active beamforming and scheduling of the satellites and the phase shifts of RIS under the transmit power constraint for each satellite, the elevation angle constraints, and the phase shift constraints of the RIS. To handle this non-convex and NP-hard optimization problem , we propose two efficient algorithms, i.e., the Dinkelbach-BigM-Successive-Penalty (DBSP) algorithm and the Lagrangian Dual Majorization (LDM) algorithm. The DBSP algorithm is based on the alternating optimization approach, which can effectively solve the formulated non-convex optimization problem with multiple dual and complex optimization variables. Specifically , we first employ the Dinkelbach method, successive convex approximation, big-M formulation, and semidefinite relaxation method to optimize the active beamforming and the scheduling of the satellites. In addition, the penalty convex-concave procedure approach is utilized to design the phase shifts of RIS. To reduce the complexity and improve computational efficiency, we propose the LDM algorithm and derive an analytical solution for active beamforming and phase shifts by exploiting the Lagrangian dual transform, quadratic transform, and majorization-minimization algorithms. Numerical simulations are conducted to demonstrate the efficiency and convergence behavior of the proposed algorithms. Moreover, it is also demonstrated that the proposed algorithms are superior to other benchmarks, corroborating the benefits of deploying an RIS in the multi-satellite network. Index Terms—Reconfigurable intelligent surface, multi-satellite IoT networks, resource scheduling, energy efficiency, mixed-integer nonlinear programming, and quadratic transform.