The shallow shear-wave velocity structure obtained from surface-wave dispersion inversion is essential for seismic site characterization; however, deterministic inversion provides only point estimates without uncertainty, while existing probabilistic approaches often lack coverage calibration, systematic robustness evaluation, and physical-consistency checks. A mixture density network is trained on a large-scale public dispersion-inversion benchmark to output a depth-wise probability distribution of shear-wave velocity Vs; conformal calibration is introduced to provide coverage guarantees, an augmentation-calibrated robust variant handles distribution shift, and dispersion forward modeling examines the physical consistency of predictions. On the test set, the shear-wave velocity prediction attains a coefficient of determination R2 of 0.904 and a root-mean-square error of 0.199 km/s; this calibration corrects the empirical coverage of the nominal 90% interval from an over-covered 0.950 to a precise 0.903 while narrowing the interval width by 14.3%. Under observational noise and out-of-distribution geology, the standard conformal coverage degrades, decreasing to 0.829 for faulted sites, and augmented calibration partially restores it to 0.868; a contrast between generic and frequency-dependent physics-guided perturbations yields nearly identical coverage, indicating that the recovery stems mainly from augmentation breadth rather than perturbation spectral shape, with residual under-coverage remaining under severe shift. A dispersion-consistency diagnostic reveals a positive correlation (0.558) between physical residual and predictive uncertainty, showing that the uncertainty captures physical inconsistency. The framework delivers calibrated, robust, and physically consistent uncertainty quantification for probabilistic surface-wave dispersion inversion.