Large-scale end-to-end models such as Whisper have shown strong performance on diverse speech tasks, but their internal behavior on pathological speech remains poorly understood. Understanding how dysarthric speech is represented across layers is critical for building reliable and explainable clinical assessment tools. This study probes the Whisper-Medium model encoder for dysarthric speech for detection and assessment (i.e., severity classification). We evaluate layer-wise embeddings with a linear classifier under both single-task and multi-task settings, and complement these results with Silhouette scores and mutual information to provide perspectives on layer informativeness. To examine adaptability, we repeat the analysis after fine-tuning Whisper on a dysarthric speech recognition task. Across metrics, the mid-level encoder layers (13-15) emerge as most informative, while fine-tuning induces only modest changes. The findings improve the interpretability of Whisper's embeddings and highlight the potential of probing analyses to guide the use of large-scale pretrained models for pathological speech.