This paper investigates a low Earth orbit (LEO) satellite communication system enhanced by an active stacked intelligent metasurface (ASIM), mounted on the backplate of the satellite solar panels to efficiently utilize limited onboard space and reduce the main satellite power amplifier requirements. The system serves multiple ground users via rate-splitting multiple access (RSMA) and IoT devices through a symbiotic radio network. Multi-layer sequential processing in the ASIM improves effective channel gains and suppresses inter-user interference, outperforming active RIS and beyond-diagonal RIS designs. Three optimization approaches are evaluated: block coordinate descent with successive convex approximation (BCD-SCA), model-assisted multi-agent constraint soft actor-critic (MA-CSAC), and multi-constraint proximal policy optimization (MCPPO). Simulation results show that BCD-SCA converges fast and stably in convex scenarios without learning, MCPPO achieves rapid initial convergence with moderate stability, and MA-CSAC attains the highest long-term spectral and energy efficiency in large-scale networks. Energy-spectral efficiency trade-offs are analyzed for different ASIM elements, satellite antennas, and transmit power. Overall, the study demonstrates that integrating multi-layer ASIM with suitable optimization algorithms offers a scalable, energy-efficient, and high-performance solution for next-generation LEO satellite communications.