There was no marker for the quest [Stiff Negotiations] but the quest was available.

https://www.wowhead.com/wotlk/quest=12112/stiff-negotiations

v7.3.1
WotLK classic

When does this quest become available, after Narf's chain? I did it myself but didn't notice a missing pickup since I accepted it immediately.

It was available before I've done Nozzlerust Post quest, before even the breadcrumb from the dead goblin. It's probably wrongly marked as part of the chain since Nozzlerust Post quests naturally lead you to it.

Are you absolutely sure this quest doesn't require both Lumber Hack and Harp On This! in order to be picked up? There's a plethora of Wowhead comments, including ones from WotLK Classic, suggesting as such.

I think the lack of map marker for you may have been a one-off race condition.

It's been a while, but I do clearly remember passing over The Crystal Vice and spotting the quest marker before I even picked up the quest from dead goblin. I didn't take it because I always do quests in their natural order if I can help it.
There is a comment on NPC's page that suggests I'm right:

vexis58 on 2008/12/08 (Patch 3.0.3)

I wandered into it on my way to Fordragon Hold. Completed the Might of Dragonblight achievement without ever finding a goto quest for this hub.

@Yttrium-tYcLief what is your thought on this?