Massive random access is an important technology for achieving ultra-massive connectivity in next-generation wireless communication systems. It aims to address key challenges during the initial access phase, including active user detection (AUD), channel estimation (CE), and data detection (DD). This paper examines massive access in massive multiple-input multiple-output (MIMO) systems, where deep learning is used to tackle the challenging AUD, CE, and DD functions. First, we introduce a Transformer-AUD scheme tailored for variable pilot-length access. This approach integrates pilot length information and a spatial correlation module into a Transformer-based detector, enabling a single model to generalize across various pilot lengths and antenna numbers. Next, we propose a generative diffusion model (GDM)-driven iterative CE and DD framework. The GDM employs a score function to capture the posterior distributions of massive MIMO channels and data symbols. Part of the score function is learned from the channel dataset via neural networks, while the remaining score component is derived in a closed form by applying the symbol prior constellation distribution and known transmission model. Utilizing these posterior scores, we design an asynchronous alternating CE and DD framework that employs a predictor-corrector sampling technique to iteratively generate channel estimation and data detection results during the reverse diffusion process. Simulation results demonstrate that our proposed approaches significantly outperform baseline methods with respect to AUD, CE, and DD.