I complete Quests (or Rumors, as they are apparently called now[?]) by using the "Bring up specific event or rumor" to people in civilizations right? I wasn't sure if I'm doing it right, since my events don't disappear from the list, and it's full of 8 pages of inconsequential stuff revolving around animal attacks and such, so finding the actual important stuff is difficult.
And do I need to say, specifically bring up the death of the Bandit's/Vermin's leader to complete a quest, or does my fame go up for every report of each individual Bandit's death?
Quests have been no longer a thing for DF204 in general, sadly. But your fame will go up accordingly as you mention tales of your exploits. At first only the person you tell it to regards you favorably, but given about a week or two the rumor spreads and affects the fame of other NPCs.
I still wish there was a way to implement DF2012-style quests, or at least a way to track WHICH troubles you have personally decided to resolve.
In addition, it'd be nice if those troubles were better acknowledged when you take the appropriate action to neutralize them.
This leads ME to a question now. In the case of megabeasts (the only trouble event that can be conclusively resolved with just 1 kill), if you kill a megabeast and allow the rumor of its death to spread, will that "this thing exists and is annoying" event eventually disappear from an NPC's troubles list?