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Robust depth completion is fundamental to real-world 3D scene understanding, yet existing RGB-LiDAR fusion methods degrade significantly under adverse weather, where both camera images and LiDAR measurements suffer from weather-induced corruption. In this paper, we introduce AURORA-KITTI, the first large-scale multi-modal, multi-weather benchmark for robust depth completion in the wild. We further formulate Depth Completion and Denoising (DCD) as a unified task that jointly reconstructs a dense depth map from corrupted sparse inputs while suppressing weather-induced noise. AURORA-KITTI contains over \textit{82K} weather-consistent RGBL pairs with metric depth ground truth, spanning diverse weather types, three severity levels, day and night scenes, paired clean references, lens occlusion conditions, and textual descriptions. Moreover, we introduce DDCD, an efficient distillation-based baseline that leverages depth foundation models to inject clean structural priors into in-the-wild DCD training. DDCD achieves state-of-the-art performance on AURORA-KITTI and the real-world DENSE dataset while maintaining efficiency. Notably, our results further show that weather-aware, physically consistent data contributes more to robustness than architectural modifications alone. Data and code will be released upon publication.