Serverless computing is increasingly used for data-processing applications in both science and business domains. At the core of serverless data-processing systems is the scheduler, which ensures dynamic decisions about task and data placement. Due to the variety of user, cluster, and workload properties, the design space for high-performance and cost-effective scheduling architectures and mechanisms is vast. The large design space is difficult to explore and characterize. To help the system designer disentangle this complexity, we present ExDe, a framework to systematically explore the design space of scheduling architectures and mechanisms. The framework includes a conceptual model and a simulator to assist in design space exploration. We use the framework, and real-world workloads, to characterize the performance of three scheduling architectures and two mechanisms. Our framework is open-source software available on Zenodo.