Dist of Columbia
How should this be done? No. Any way you like is fine, but I prefer flipping the entire stack over all at the same time.
Is there anything in the rules specifying it?
One at a time to make it exciting? And if so, if you see one fail do you go on revealing the others? Yes.
Also, 5 consecutive rejected teams loses the entire game or just the mission? The entire game. But the way I play it (when you don't have plot cards at least) is no one votes on the .5 mission proposal, you just go on it after the leader picks its members. It's just easier that way.