Genghis was an avid user of both established and novel siege techniques, and made great use of captured Chinese engineers. Though gunpowder was not core in his siege techniques, his successors utilized it to a far greater extent.

The Great Wall was never meant to decisively stop large nomad forces, only to stall them long enough for an army to be mustered instead of facing a bunch of nomads at your capital around the same time you hear about them crossing the border. Gates can be battered down, sections of wall can be scaled or undermined, etc.

(smaller nomad raids it was expected to stop)

The Mongols had the production centers of everyone they conquered. Also this is pre industrial, producing explosives of the era didn't require heavy machinery it required knowledge and materials, and a bunch of nomadic horse archers are going to produce one hell of a supply line.

Think of the Mongols as vikings with horses instead of boats.