How does local trouble effect heroes such as Dain and Gimli with ongoing abilities? Will they constantly trigger the card text for a threat increase?
Those are constant abilities and are never triggered. Only Action and Response abilities are triggered.
Approaching Cair Andros says if any players threat is 37 or higher it gains siege. Is this just checked when the quest is flipped or does it gain siege at anytime a players threat reaches 37 or higher?
The effect is not a "When revealed" effect, so it triggers at any point any player's threat is greater than or equal to 37.