The beast plague is something that dwells within humanity to begin with, and beasts are simply a thing that happen from time to time (Constable gear and Yamamura's Hunter gear both mention beasts from elsewhere) but the reason that the beast plague is such a huge issue in Yharnam is because they use so much of the old blood, which greatly increases the risk. Mergo is not related to the beastly scourge, at all.
No. Each hunt is different. Hunts have gone on since the days of Pthumeru, long before the school of Mensis was around to muck things up, long before Rom ascended to become a Great One. The only reason Mergo is the goal is because Mensis used his umbilical cord to commune with him, specifically so that a hunt would occur.
Hunts coincide with the appearance of special children. It is a relationship of both cause (Mergo appears, the hunt begins) and effect (the hunt begins, Arianna is impregnated). Mergo is special simply because he is the trigger for this hunt. The moon presence's goal is for us to kill Mergo. Why that is can only be theorized at.
The notes you mention tell us everything here: the moon presence, taking the form of the blood moon, is lured by Mensis by using Mergo as bait. This triggers the hunt, and the bad effects associated with it. There are two types of hunt: a relatively common occurrence where people go out and slaughter beasts to keep their population down, and rarer hunts when the blood moon appears and everything falls apart. The blood moon excites the blood, causing rampant insanity, beasthood, and allowing Oedon to have his liaison with Arianna. The only way to truly end the plague of beasts is to remove the source, which is beasthood at the core of our human natures. The only way to save Yharnam from collapse in a single night is to prevent the blood moon from rising. The only way to save Yharnam in the long run is to stop the usage of the old blood. None of those happen on our hunt, unless you take the true ending and kill the Moon Presence, which at least will prevent other hunts like ours. That is the one thing we can do that is different from the rest of the hundreds of hunters who preceeded us. If we take this route, paleblood hunts will no longer occur. If we accept Gehrman's mercy, nothing changes, because it is the same outcome that has happened a hundred times before. If we replace Gehrman, the only change is the identity of the hunter's helper in the hunts that have yet to come.