The growing adoption of deep learning (DL) for chest X-ray (CXR) diagnosis faces three significant barriers that this study addresses. First, inherent ambiguities in CXRs - particularly overlapping tissue intensities between pathological and healthy regions, along with common noise artifacts - create indeterminate zones where conventional DL models frequently err. Second, the scarcity of high-quality annotated datasets and persistent class imbalance problems lead to biased and overfitted models. Third, the opaque decision-making process of DL systems undermines clinical trust, especially in borderline cases. To resolve these challenges, we implement Neutrosophic Sets (NS) to explicitly quantify and manage uncertainty at the pixel level through truth-falsity-indeterminacy memberships, particularly effective in clarifying ambiguous infection boundaries. Simultaneously, we employ radiologist-validated Data Augmentation (DA) techniques to mitigate data scarcity and imbalance issues. Our results demonstrate NS filtering enhances model reliability, improving EfficientNetB0 accuracy by 3.13% (94.79% to 97.92%) in uncertain regions, while DA boosts MobileNetV3Large's generalization capability with a 5.78% accuracy gain (93.75% to 99.53%). Building on these findings, we propose MV3Lung-NS, an integrated framework combining NS preprocessing, DA, and MobileNetV3Large that achieves state-of-the-art performance (99.53% accuracy, 99.65% precision) on pulmonary infection diagnosis. To bridge the interpretability gap, we implement Explainable AI (XAI) methods including SHapley Additive exPlanations (SHAP), Local Interpretable Model-agnostic Explanations (LIME), and Gradient-weighted Class Activation Mapping (Grad-CAM), providing visual evidence that model decisions align with radiological markers of infection. This work makes dual contributions: advancing neutrosophic theory through empirical validation in medical imaging and delivering a clinically viable solution that addresses both technical and trust-related barriers in AI-assisted diagnosis.