Hypergraph-based modeling has gained significant attention for capturing complex higher-order interactions among vertices. While random walks serve as fundamental tools for analyzing hypergraphs, existing approaches either fail to fully leverage edge-dependent vertex weights (EDVWs) or lack sufficient expressiveness to model intricate hypergraph structures. To address these limitations, we propose a unified random walk framework that integrates hyperedge degrees and vertex weights, offering a more robust approach to hypergraph modeling. We establish equivalence conditions between hypergraph and graph random walks, leading to a novel unified random-walk-based hypergraph Laplacian that incorporates EDVWs, ensuring expressiveness and desirable spectral properties. Building on this foundation, we introduce the General Hypergraph Spectral Convolution (GHSC) framework, which extends existing Graph Convolutional Neural Networks (GCNNs) for effective hypergraph learning, supporting both edge-independent and edge-dependent vertex weights. Extensive experiments across diverse datasets, including citation networks, visual objects, and protein modeling tasks, demonstrate state-of-the-art performance, with notable improvements in protein structure modeling using EDVW-hypergraphs. This work advances the theoretical understanding of hypergraph random walks and spectral theory while providing a versatile framework for deep hypergraph learning. Code is available at GHSC_H_GNNs.