In recent years, Mobile Edge Computing (MEC) has been widely used for latency-sensitive tasks, but task scheduling in dynamic edge environments still faces two key challenges. First, edge devices are prone to failures, and existing fault-tolerance mechanisms lack task-aware modeling, making it hard to ensure timeliness and reliability under failures. Second, due to limited perception, high communication costs, and complex task structures, current scheduling strategies still struggle with adaptability and stability in dynamic systems. In this paper, we propose a Fault-Tolerant Discrete Soft Actor-Critic scheduling algorithm (FT-DSAC). Initially, we design a Primary-Backup-based Fault-Tolerant (PBFT) scheduling mechanism, which constrains task offloading locations and start times to effectively mitigate the impact of failures on task execution. Furthermore, we incorporate the Centralized Training and Distributed Execution (CTDE) architecture, which enables implicit collaborative scheduling decisions among edge servers to optimize system performance and reduce communication overhead. Finally, We conduct extensive experiments using both simulated data generated by DAGGEN and real-world workflow data. Experimental results show that the proposed algorithm significantly improves task execution success rates by 6%-19% and reduces latency by 9%-27% compared to mainstream benchmarks.