According to the novels, greyscale is not a simple disease, but it's not tied to The Others. Rather, it's tied to an historical figure known as Garin the Great, a Rhoynish prince who lived about 1,000 years ago.
NOTE: Minor spoilers for A Dance with Dragons -- we find all of this out in that novel, but I'm not including any real plot points.
During the so-called Second Spice War, a small group of Valyrian lords allied with Volantis to attack the Rhoynar. Prince Garin led the resistance, and initially was winning, until Valyria decided to throw all-in and send an army of dragons. The Valyrian army crush the Rhoynar and captured Garin.
According to legend, the captive Garin called out to the goddess of the river, Mother Rhoyne, and she made the rivers rise up out of their banks into a massive fog, and when the fog had lifted, the Valyrians were all dead.
The area of the river where this supposedly happens—The Sorrows—is still considered cursed, and the greyscale sufferers still live there. The same legend claims that Garin is actually still alive, and that he is the Shrouded Lord that rules the Sorrows. It also claims that "Garin's Curse", which killed the Valyrians, was actually greyscale, brought on by Mother Rhoyne.
The idea that it started with Garin is reinforced a bit by another legend about the Shrouded Lord: that he was originally a statue, come to life by the kiss of a strange woman. If Garin was the first person afflicted with greyscale, he would definitely have looked like a statue come to life to people that had never seen it before.
Of course, these are all just legends, and even Garin's legend has a touch of Other in it (specifically, the woman who brought the Shrouded Lord to life was "grey" with "lips as cold as ice". ) The real origins of greyscale have never been explored, and records of it are sketchy, so it's not clear how recently the disease appeared, or if Garin could really have been the origins.
To address the points in your question directly:
- People with grayscale have cracked skin that's almost black, very different from the way Others are described.
- Greyscale thrives in damp climates, like the rivers.
- I have no idea where reddit came up with the "fact" that greyscale is tied to the Grey Waste; it's tied to the River Rhoyne and The Sorrows
- "What is dead may never die, but rises again, harder and stronger" is a saying of the Iron Island, and has nothing to do with Essos. (The Rhoynish that fled Essos ended up in Dorne.)