Answers are in FAA AC120-28D
The answers to virtually all these Cat III, Cat IIIA and Cat IIIB, and DH and No-DH questions are contained in FAA AC120-28D. There are some more recent minor exceptions and subtle twists, per issuance of more recent Op-Specs, such as for Alaska Airlines' Hybrid Cat III Ops, using both autoland, and the HUD AIII Mode to monitor the AP.