Accurate and interpretable classification of infant cry paralinguistics is essential for early detection of neonatal distress and clinical decision support. However, many existing deep learning methods rely on correlation-driven acoustic representations, which makes them vulnerable to noise, spurious cues, and domain shifts across recording environments. We propose DACH-TIC, a Domain-Agnostic Causal-Aware Hierarchical Audio Transformer for robust infant cry classification. The model integrates causal attention, hierarchical representation learning, multi-task supervision, and adversarial domain generalization within a unified framework. DACH-TIC employs a structured transformer backbone with local token-level and global semantic encoders, augmented by causal attention masking and controlled perturbation training to approximate counterfactual acoustic variations. A domain-adversarial objective promotes environment-invariant representations, while multi-task learning jointly optimizes cry type recognition, distress intensity estimation, and causal relevance prediction. The model is evaluated on the Baby Chillanto and Donate-a-Cry datasets, with ESC-50 environmental noise overlays for domain augmentation. Experimental results show that DACH-TIC outperforms state-of-the-art baselines, including HTS-AT and SE-ResNet Transformer, achieving improvements of 2.6 percent in accuracy and 2.2 points in macro-F1 score, alongside enhanced causal fidelity. The model generalizes effectively to unseen acoustic environments, with a domain performance gap of only 2.4 percent, demonstrating its suitability for real-world neonatal acoustic monitoring systems.