Also, yes, if you manage to end a turn with no obstacles in play in Extraction (or any mission), you get to remove the Crossfire card (and you don't discard it, and you don't draw a new one).
Reviving an "old" thread (lol) sorry, but I just thought of a situation in the Extraction mission... Maybe I am thinking too much or i am forgetting a rule.

It's a 2+ runners game. Let's say that the starting player has really good cards/luck at the beginning of a round and that he defeats all obstacles on his turn during an Extraction. So, we remove the Crossfire event (put it in the bottom of the deck). Now, we follow the instructions for the next round, as indicated on the Client card.
But which player begins the next round in that situation?
- The starting player, since this is a new round according to the Client card terminology - this way, we have to draw a new Crossfire event.
Or
- The player at the left of the starting player, since he would be the next one to play after the Starting player ended the previous round by defeating everyone (as in a Crossfire mission) - this way, we do not draw a new Crossfire event until the turn of the starting player.
If the first option is the good one, that means that a quick victory in a round in Extraction is a little less rewarding (but still a great thing, since we keep low the crossfire leve!) than in Crossfire mission.