My question is regarding the instruction for many quest that say "add to the staging area".

Do any effects trigger on a card, say a monster card that has Doom:2 or Hide: when the instruction is at the start of a quest? I understand that some quests word it differently, for example it says "draw X cards for each player". Which then triggers effects as normal, but many quests have you adding specific locations or monsters to the staging area, some of which have "When Revealed" effects or keyword effects as mentioned (Doom 2 for example).

If a card is merely "added" to the staging area, then it does not count as being Revealed. Therefore its When Revealed effects do not trigger. Any Forced effects on the card of the form "When this card enters play" or similar will trigger. Whether a particular keyword triggers or not will be defined in the rules of the keyword: does it refer to being revealed or not? The standard keywords Doomed, Surge, Guarded all specifically reference being revealed, so none of these trigger when a card is merely added to the staging area.