Document-level event extraction (DEE) aims to extract the structured event information from a given document, facing two critical challenges: (1) event arguments always scatter across sentences (arguments-scattering); (2) multiple events can co-occur in one document (multi-event). Most recent studies mainly follow two simplified settings to ease the challenges: one simplifies DEE with the no-trigger-words design (NDEE), and the other focuses on event argument extraction (DEAE), a sub-task of DEE. However, the former excludes trigger extraction and suffers from error propagation in the sub-tasks. The latter relies heavily on the gold triggers as prerequisites and struggles to distinguish multiple arguments playing the same role in different events. To address the limitations above, we propose a novel joint trigger and argument extraction paradigm SEELE to enhance the DEE model via incorporating SchEma-awarE descriptions into Document-Level Event extraction. Specifically, the schema-aware descriptions are leveraged from two aspects: (1) guiding the attention mechanism among event-aware tokens across sentences, which relieves arguments-scattering without error propagation; (2) performing the fine-grained contrastive learning to distinguish different events, which mitigates multi-event without gold triggers. Extensive experiments show the superiority of SEELE, achieving notable improvements (2.1% to 9.7% F1) on three NDEE datasets and competitive performance on two DEAE datasets. Our code is available at https://github.com/TheoryRhapsody/SEELE.