You'd likely get to make a save if you roll that 1 to catch yourself, maybe for less damage, but yes, the change to fall and potentially die, however small, is there, provided the fall is from a fatal height.
In the games you play, they do. In the games I play they (by and large) do not. Both games are fine choices, but they are quite different in many ways.
None of which really appeal to me. Does the fact that it influenced the creation of other games mean I have to follow its lead on RPG discussion and terms?
If I tried to use them, they definitely would be. In every Narrativist I've ever played in, they were used in an inartful way, playing up the parts of those mechanic I dislike.
You could fail to climb it. You could conceivably injure yourself (I've certainly done that trying to climb a wall). It is inherently an at least moderately dangerous activity. There's a reason right there. The stakes don't have to be unconnected to physically climbing the wall, and they don't...
Because the roll is about trying to climb a wall (presumably using a metric related to climbing), not why you're climbing a wall. That's a separate roll if it's a roll at all. If you make a roll to see if you successfully make a sword in your forge, what would your reason for making that sword...