"The noose is hanging" is a strange phrase that I've never heard, but it could perhaps be used properly given the proper context. Note that this context is almost certainly never literal. I'd imagine it used in the context where a trap has been set for someone, but not usually one which results in their death.
One of the examples for it sounds very strange to me:
I prepped the lab for the experiment, so the noose is hanging, whenever you want to begin.
I can't imagine anyone saying this in this context. This sounds like an antiquated phrase.
As for "put (one's) head in a noose" and "stick your head in a noose," both of the example sentences you link sound very strange. I'd contend that in modern US English, most people would use "shot in the foot."
So instead of:
He had a real shot at winning the election, but he put his head in the noose with such inflammatory remarks.
He had a real shot at winning the election, but he shot himself in the foot with such inflammatory remarks.
This phrase is meant to convey carelessness on the subject's part that inflicts damage to them.
Note that in the original, it actually said "put his head in the noose in the foot" (which is nonsense). This suggests that at one point the phrase was (or was supposed to be) "shot himself in the foot" anyway.
For the second example, I'd use "stick their neck out" instead of "put their head in the noose". So instead of:
Everyone knew the idea was bad, but no one was willing to put their head in the noose and criticize it during the meeting.
Everyone knew the idea was bad, but no one was willing to stick their neck out and criticize it during the meeting.
Here the phrase does somewhat evoke imagery of executions. It (hyperbolically) suggests that someone would be willing to risk being executed by the guillotine to do something. Although, everyday usage is never that literal.
Hopefully, you see the pattern here that phrases like this are rarely used that literally.
For your context (and given your other question about words surrounding a suicide), I'd suggest the phrase "talked down from the ledge" to indicate saving someone from a suicide.
If the near death incident is some sort of bodily harm (particularly an illness), I'd use "nursed back to health:"
The flu nearly killed the child, but her parents nursed her back to health.
You may also use the phrase "guardian angel" to describe such a person who saves someone from near death.
But, I don't think you'll come across many phrases that uses "noose" so literally.