In the rule book, I found that the keyword "surge" is explained like"when an encounter card with the surge keyword is revealed during the staging step of the quest phase, reveal 1 additional card from the encounter deck . Resolve the surge keyword immediately after resolving any when revealed effects on the card."

So I wonder if we need to add the card been revealed by the surge keyword into the staging area? or just trigger its keyword and "when revealed" effect and put it back to the encounter deck? because it doesn't say "reveal and add into the staging area".

It´s added to the staging area

If a treachery card has surge and the when revealed is canceled by test of will, does surge still trigger? Same question for doomed keywords. I've been playing that these keywords do trigger when canceled, but now I'm not sure.

Surge and Doomed still trigger -- A Test of Will and Eleanor only cancel the 'when revealed' effects of cards, the Surge/Doomed part is a separate effect.