Speaking about WoW specifically, since I'm not familiar with the others, I've always been curious about their quest system. Specifically keeping track of what are available for the character efficiently along with the event system to flag quests as completed, etc. There's so many of them. I have to assume they're spatially limited. You enter a zone, or an area, and the system loads up all of the quests located in tha…
In short, from what I know, yes, quests are stored in "quest log" fields of character data in server DB, and they are tracked by the clients and checked by the server. Some simple auto quests like "find this item" are not even tracked by server and only stored on completion. Since both client and server have all game data, the client knows about all possible quests and only shows to the player what is appropriate at a current state.