Disclaimer: The developers have called Dark Souls a spiritual successor to Demon's Souls, but also asserted that it is not a direct sequel or even takes place in the same universe.
One time when the Old One woke up, humans failed to lull it back to slumber, and the colorless fog spread out over the entire world, which started the age of ancients in Dark Souls, with the soul-starved humans hiding in the dark. So, what happened to the old one? Perhaps the Old One died somehow, and its death started the fire that had the Lord Souls.
After the fire came, the colorless fog from the age of ancients left. Ithe fog was caused by the Old One, the only reason that would happen is because it wasn't around anymore. But, there are still some remnants of the fog, the white fog at boss gates. This fog could be a lesser version of the fog that the Old One emitted by the bosses. When the boss is a demon it would emit the fog, but only enough to cover its immediate surroundings. As for human enemies like Ornstein, they became demons similar to your character in Demon's Souls. They absorbed souls and became stronger to the point that they are now demons.
The Old One's death caused the fire because of the relationship fire and souls have. The fire is going out during the time you play Dark Souls, and one option you have is to sacrifice yourself so that your soul can fuel the fire for a short time, but the Old One's soul may have been powerful enough to create the First Flame.
When the Witch of Izalith's soul became chaotic, maybe it created more demons from that same fire. However, he demons from Izalith are not like the demons from Demon's Souls. They don't devour souls for nourishment, or if they do it's just like anyone else in Dark Souls and not enough to identify them, since everyone else does it. They're just a different and chaotic form of life. The witch of Izalith did not make demons again, since the Old One's demons are extinct and forgotten.