Does everyone in that party need to make the CON test when you force march, or just the leader?
I believe each adventurer who Force Marches must make the check. (I don't think it would make sense for it to be just the leader.)
Assuming some succeed, and others fail, does this mean the whole party can no longer Force March until those who failed have rested?
Right, although the people who can still Force March could take off next turn, splitting the party, while those who can't Force March rest or walk slowly...