so you finish the encounter, choose the mission as the reward, then you do speed checks or fight a bunch of things that are listed on the mission right?
also on the final mission cards, we got "the last fragment" so after rolls the party had to last 2 turns and the party beat it twice, that would mean that group wins right.
For the crisis, it counts down as you search for the required location too right? Do you still fight the crisis even if you didn't manage to find the location?
Yes Indeed. Finish the Encounter, choose the Mission, then fulfill whatever requirements are on the Mission card.
Yes, that would mean that party won. (As long as they fought him over 2 consecutive turns, not just fighting him twice during one turn.)
Yes, the Crisis counts down after every Encounter phase. If the timer counts down and the required location hasn't been found, everyone that has completed at least one Basic Mission has one chance to defeat the Crisis. (Also, the Final Mission usually comes up late in the game. I would suggest looking through the discard pile just to make sure it hasn't already been in play)