Millimeter-wave massive multiple-input multiple-output systems employ highly directional beamforming to overcome severe path loss, and their performance critically depends on accurate beam alignment. Conventional codebook-based methods offer low training overhead but suffer from limited angular resolution and sensitivity to hardware impairments. To address these challenges, we propose a deep learning-enhanced super-resolution beam alignment framework with three key components. First, we design the Quaternary Search-based Super-Resolution (QSSR) algorithm, which leverages the monotonic power ratio property between two discrete Fourier transform (DFT) codebook beams to achieve super-resolution angle estimation without increasing measurement complexity relative to binary search. Second, we develop QSSR-Net, a gated recurrent unit-based neural network that exploits sequential multi-layer beam measurements to capture angular dependencies, thereby improving estimation accuracy, robustness to noise, and generalization across diverse propagation environments. Third, to mitigate the adverse effects of hardware impairments such as antenna position and phase errors, we propose a parametric self-calibration method that requires no additional hardware overhead and adapts compensation parameters in real time. Simulation results show that the proposed framework consistently outperforms binary search and even exhaustive search at high signal-to-noise ratios, achieving substantial performance gains while maintaining low overhead.