Federated learning (FL) is a powerful technology that enables collaborative training of machine learning models without sharing private data among clients. The fundamental challenge in FL lies in learning over extremely heterogeneous data distributions, device capacities, and device state availabilities, all of which adversely impact performance and communication efficiency. While data heterogeneity has been well-studied in the literature, this paper introduces FLHetBench, the first FL benchmark targeted toward understanding device and state heterogeneity. FLHetBench comprises two new sampling methods to generate real-world device and state databases with varying heterogeneity and new metrics for quantifying the success of FL methods under these real-world constraints. Using FLHetBench, we conduct a comprehensive evaluation of existing methods and find that they struggle under these settings, which inspires us to propose BiasPrompt+, a new method employing staleness-aware aggregation and fast weights to tackle these new heterogeneity challenges. Experiments on various FL tasks and datasets validate the effectiveness of our BiasPrompt+ method and highlight the value of FLHetBench in fostering the development of more efficient and robust FL solutions under real-world device and state constraints.