It is unrelated to 1) quality of the safe 2) skill of the safe opener. It is not unrelated to the motives of the characters, but neither of the things you derive your odds for the paper being there measures that. So yes, these things are unrelated.
Right. So you admit that you shouldn't let the person with the low skill to try to open the safe as that will crash the odds of the paper being there?
And I get the idea of the motives of characters and desires of the players shaping the reality just fine, it just seems incoherent tome to bring obviously simulationist measures of safe quality and lockpicking skill into it then.
I don't "admit" anything. I have a player who, in both Burning Wheel and Torchbearer, likes to insert his PC into situations to which the character is not especially well-suited, in order to get "ticks" for progression on his character's abilities. Or on one occasion, when the stakes were very high, he insisted on going from A to B via the catacombs rather than above ground because he wanted to get a Catacombs-wise check out of it.
So it's completely feasible that a person with low skill might try to open the safe. But the stakes of that would reflect
their PC's Beliefs, not the Beliefs of some other PC. And its a premise of the game that that PC's Beliefs are as important to that player as the Beliefs of the skilled safe cracking PC are to
that PC's player.
If it's important to the non-safe cracking player that the dirt be found, then presumably they will be trying to do that through their preferred method. Perhaps the PCs already had an argument about that, and the safe cracker won the Duel of Wits -
We're going to try the safe first, and I'll be the one to open it!.
As far as lockpicking skill and safe strength are concerned, in Burning Wheel these are not simulationist things. I think it was somewhere way upthread that I posted that BW has a veneer of process simulation but - in virtue of its core principles for resolution which include "say 'yes' or roll the dice", "intent and task", and "let it ride" - is actually a "story now" system.
What the lockpicking skill and safe strength do is provide colour. This is discussed briefly in the core rules but at length in the Adventure Burner (and that discussion is reproduced in the Codex): the imagined world is given shape and texture by the obstacles assigned to different components of it. So if you want the colour of your PC to be
someone who collects dirt by finding it in safes, you better start building up your lockpicking skill! Maybe do some practice - but that will trigger seasonal Resources checks - and that is what drives the game.
I keep getting told that can be done but no explanation of what actually doing any of this looks like in the scenario I described.
I gave some descriptions in the post you're replying to!
Players are both around safe.
1. Who makes the first move?
This might already be settled by some prior consequence. If it's an open question, we can test Speed vs Speed.
2. The first player wants to open the safe and find incriminating evidence of his arch nemesis. What move does he make?
I'm talking Burning Wheel. So the move will be a test of some salient skill. What that is depends on both the task and the intent declared by the player. Perhaps lockpicking, perhaps augmented by other relevant skills (eg Safes-wise).
3. Does that move resolve to completion with out player B being able to offer any input? Or does player B get to interrupt the move or declare his own move before it’s resolved?
Burning Wheel doesn't have an action economy in the sense your question seems to presuppose. What consequence flowed from player B losing the Speed check? The GM will have narrated that having regard to what is at stake for each PC, which will depend on the Beliefs of the two PCs. And that will have been clear prior to resolving the Speed check (so that the players know how many of their resources - Fate or Persona points, Call-On traits, etc - they might want to use on the check).
The last time something in this neighbourhood happened in my play, two PCs drugged a NPC so they could get to a different NPC's home, where he was resting from severe wounds, without the drugged NPC, who wanted to kill that other NPC, following them. But then (as I said above) they went via the catacombs, and failed the Catacombs-wise check. So the next scene was the PCs, stuck in a dead end beneath a street grate not knowing which way to go, with the no-longer-drugged NPC looking mockingly down at them. So that made it a race: Speed vs Speed. The NPC won, and so the next scene was the PCs racing into the resting place just in time to see the NPC having beaten them there, about to decapitate the resting NPC. The NPC's Sword check succeeded, and so the resting NPC's head went rolling.
A race for the safe, and the establishment of consequences and subsequent framing, doesn't strike me as being any harder to do.
3b. In what order are moves resolved? Is there the streetwise or battle of wits resolution first and then the player that wins that gets first crack at the safe? Or does it work in a different order?
It depends on how scenes are framed and consequences incorporated. I've given an example above of how a Duel of Wits might be relevant in the safe context.