Does the -OnTriggerStay collider- script have to be child of collider triggering the function?

Or can you tell it, use a different objects collider as trigger?

No, it has to be on the object which is colliding.

You’re free to send messages from the OnTriggerStay function to whatever objects need to know about the collision.

You can even call other OnTriggerStay functions if you need to for some reason, though that might get confusing.

I don’t know if this is exactly what you’re asking, but all OnTrigger events are sent to scripts attached to the trigger object and to the object that entered the trigger, not to their parents or children. If you have a script attached to object A and want to detect when something entered the trigger in a child of A, a simple solution is to use SendMessageUpwards to make the trigger call a function in its parent - add the code below to the child trigger script:

function OnTriggerEnter(other: Collider){
  SendMessageUpwards("TriggerEntered", other);
}

Define the function TriggerEntered(other) in the parent object’s script:

function TriggerEntered(other: Collider){
  // the collider "other" entered the child trigger
}