The hard stops provide physical limits that prevent the filter from rotating into the cross-polarization zone, ensuring you never get a dark 'X' across your image.
No, high-quality Hard Stop ND filters use precision-ground optical glass that allows the camera's phase-detection and contrast-detection autofocus systems to function normally.
Usually no. Most Hard Stop NDs have a slightly larger front diameter to prevent vignetting, often requiring a cap one size larger (e.g., an 82mm filter may need an 86mm cap).