bangkorai - any chest is possible including delves, difficulty don't think matters, got mine on both servers on easy, one in delve one overland.
bad man's - from skewer and shalk(eu/ na), from boss for me was wild hunt.
Can confirm. Bosses drop the Wild Hunt lead and if you got the Rail instead from a boss, that's probably a bug. Trash mobs ranging from Skeevers, the insane folks down there, skeletons, shalks, and snakes all drop it (not the snake boss).
For the Signet, it's any and all chests in Bangkorai. Difficulty doesn't matter as I found it on a Master chest my first go and then another on an Easy chest. I'm not sure if the Treasure Hunter perk in CP helped my RNG with that or not, but it's something the OP may want to consider anyways since it's not a bad loot farm.
Drop chance for the Order-Etched Gallery Rail in Bad Man's Hallows is low.
I've farmed the trash mobs for hours so far and it hasn't dropped for me yet.