Just off the top of my head so I may make some errors, but here goes;
Basically, researchers at Byrgenwerth Academy found an entrance to the Chalice Dungeons, which is where they found a bunch of Pthumerian shit, learned about the Great Ones and the Old Blood. The Old Blood is just the blood of Great Ones. All the blood that people drink in Yharnam is Old Blood, which has the nice effect of healing wounds and incurable diseases, but also turns people into beasts if used a lot. There was a schism in Byrgenwerth that led to the foundation of the Healing Church by Lawrence, who wanted to use the Old Blood to help people, while Provost Willem, the head of Byrgenwerth, wanted to study the Great Ones to enact a kind of forced evolution for mankind, turning people into Great Ones.
The Night of the Hunt is a regular event (maybe each full moon, maybe annually, it's not clear) where normal citizens lock themselves away so hunters can kill all the people who have turned into beasts so the city isn't completely destroyed. At first, this was done in secret as there weren't many beasts, but as the use of Old Blood became more common, it was impossible to hide the truth from the public, so it became a regular thing in Yharnam that everyone knows about. The existence of beasts most likely became common knowledge after the Scourge outbreak in Old Yharnam, which was burned to the ground. It's still unclear if the public is aware of the connection between Old Blood and the Scourge.
The stuff about Rom, Mergo and the Moon Presence I'm less certain about. Rom is a former student of Byrgenwerth that was turned into a Kin, aka a half-Great One, and her purpose seems to be hiding the Blood Moon. Mergo is a Pthumerian - a child of Yharnam, the Pthumerian Queen. Killing the Wet Nurse doesn't free Mergo, it kills him - notice that the "NIGHTMARE SLAIN" message doesn't pop up after killing the Wet Nurse, only after Mergo stops crying. The Moon Presence is a true Great One, as it has red blood rather than the grey blood of Kin.