Goals:The goal is to extinguish all bonfires. This means you cannot light any bonfires. Kindling is questionable, as it goes against the principle, but without kindling it makes it very difficult to complete a no bonfire run.
Rules:1) No lighting any bonfires.
2) Resting at a lit bonfire is OK (I think)
3) All bonfire keepers must be killed (eventually). The one in Firelink Shrine can only be killed by Lautrec, so you must allow him to do this.
4) Collecting firekeeper souls and using them to upgrade Estus flasks might be ok.
5) Extinguishing Ash Lake and the Everlasting dragon is the main obstacle. No one knows how to do this, or if it's even possible.
Starting ClassesAs far as I know, your starting class shouldn't matter. However, personally, I think Pyromancer, Cleric and Knight don't fit as Pyromancers are pro flame, and Clerics/Knights belong to Way of White. Their mission is to kindle bonfires, not extinguish them.
I personally think Deprived makes the most sense, but Cleric would give you the easiest run with their spell of Heal.
Starting GiftI believe the Pendant or nothing makes the most sense. However, the Pendant is on Rhea and she is attempting to kindle bonfires. Perhaps the pendant is in the opposite direction.
StatsStats shouldn't make any difference.
LevelLevel shouldn't make any difference.
SinningfireLess is trying not to Sin in his runs. Perhaps just use Oswald whenever you want to switch a covenant. Personally, I don't see why you couldn't sin, since those are in the eyes of a God and we are attempting to remove the gods (or, their influence).
New GamesUser Rynn has reported strange things happening on NG7.
https://soulswiki.forumotion.com/t7909p45-extinguish-the-bonfires-no-bonfire-run#165625New Game(s) may/may not be important. There is some thought that if there was something buried so deep in the game, that it would be on NG7.