Short answers, pulled from the story.
The rex sacrorum was a senatorial priesthood known as the king of the sacred things. This office fulfilled at least some sacral duties of kingship after the overthrow of the kings during the transition to republican governance.
Scholars debate whether this office originated during the Regal period before Rome became a Republic or if Arnaldo Momigliano argued that the figure was created specifically during the formation of the Republic. The office continued until it was probably abolished by Theodosius I, with mention ending after Augustus revived the position following Antony's civil war.
During the Roman Republic, the pontifex maximus chose the rex sacrorum from a list submitted by the College of Pontiffs. A strict requirement demanded he be born from parents married through the ritual of confarreatio and enter into marriage using this same ancient form himself.
On the Kalends of every month, the rex sacrorum held a sacrifice within the Regia while performing capite velato meaning with his head covered. On the Nones, he announced the dates of festivals for the coming month to the public.
The regina presided at the sacrifice of a sow or female lamb to Juno on the Kalends of every month alongside her male partner. She wore a headdress called the arculum made from pomegranate twigs tied with white woolen thread as recorded in inscriptions like those of Sergia Paullina shortly before 112 AD.