Short answers, pulled from the story.
The earliest dated inscription using Invictus as an epithet for Sol dates to AD 158. This record appears in the Corpus Inscriptionum Latinarum alongside a dedication by a priest of Jupiter Dolichenus.
Emperor Aurelian reformed the Roman cult of Sol in AD 274 after victories in the East and elevated the sun god to one of the premier divinities of the Empire. He changed the status of priests from sacerdotes to pontifices within a new college and built a temple dedicated on the 25th of December 274.
Imperial coinage continued bearing images of Sol until 325 or 326 under Constantine. Coins struck between 274 and 275 depict Aurelian wearing a radiate crown which living emperors wore from Nero after AD 65 through Constantine.
Some scholars claim Aurelian instituted the festival Dies Natalis Solis Invicti on the 25th of December in AD 274. This date marked the winter solstice in the Roman calendar and included thirty chariot races held in Rome.
A widely held hypothesis suggests the early Church chose December 25 for Jesus Christ's birthday to appropriate the festival of Sol Invictus. The Calendar of Filocalus from 336 AD records both dates on the same day while Augustine of Hippo preached against devotees in the fifth century despite their numbers.