Short answers, pulled from the story.
The English word mystery traces its roots to the Ancient Greek plural term the Mysteries which means revealed secret. One traditional theory links it to the verb muō meaning to close or shut referring to shutting one's eyes during initiation.
The Eleusinian Mysteries lasted over a millennium until the end of the 5th century BCE. Initiates were limited to Greek speakers who had never killed anyone and possessed pure souls while gathering on the 15th of Boedromion in September or October.
Major examples include Egyptian Isis Persian Mithras Thracian Sabazius and Phrygian Cybele adopted divinities worshipped through mystery frameworks. Other groups included cults of Despoina Attis Jupiter Dolichenus Trophonius Dionysian Orphism and Serapis.
Initiation at the Samothracian Mysteries lasted from April through November peaking likely in June over two nights with large events occurring then. Future initiates entered circular spaces nine meters diameter flagstones grandstand steps listening to proclamations about absence of crime and bloodshed.
Scholarly consensus rejects direct borrowing claims despite historical attempts to tie baptism Eucharist origins to mystery religions. Arthur Nock noted near absence mystery terminology New Testament shifting attitudes as Egyptology emerged discipline.