In order to thew best of my ability:
1. Lords of cinder are those who link the fire, like the protagonist in DS1. Nobody knows how many times it has been linked since.
2. The lords are not lords of cinder, except Gwyn, I guess, because he linked his soul to it.
3. The Chosen Undead (as the protag is called in DS1) needs the souls to fuel the flame along his own. If he chooses the dark ending, he is choosing to stop them from being used on the flame.
4. The cycle happens because the fire will always fade if lit, which means someone will need to refuel it. If it is left to die, like the dark ending in DS1, the flame probably just resurfaces every once in a while.
5. King Vendrick was trying to link the flame to save his kingdom, but he was kinda fucked up by the giants. So he failed to become a Lord of Cinder. If the player went to the throne and sacrificed themselves (we don't know what would happen, as the scene ends before the choice) to link the fire, they would become one.
5. They were sucked through time and space to "recreate" the first time the flame was linked. This is why DS3 landscape is really messed up and weird. I don't know why they were chosen, maybe they were just the last five ones to do it.
7. I personally think that five are needed to complete the linking so that it will be a more permanent solution to the problem: the fire fades. Anything beyond that I don't really know.

Anyone who knows more please tell me.