The Great Sage Sun cried out in a stern, ringing voice: ‘You little ones there — be quick, go tell your cave-lord to send out my master the Tang Monk, and spare the lives of this whole cave of sprites;

Wukong at a demon's door, doing what Wukong does. 那 here is used to call out to someone -- "you little ones there." 教 is jiào, "have him send out," giving an order through someone else. 一洞 is the whole cave-ful of them, and the tone shifts to yí before the falling tone that follows. The bravado is entirely in character.

but should there grind from between your teeth so much as half a ‘No,’ I will heave your mountain domain clean over and trample your cave-mansion flat.’

牙迸半個『不』字 -- "grind out half a No between your teeth" -- is a set threat meaning the least word of refusal. The quoted 不 is read bú, its tone shifting before the falling tone of 字, which is how it would sound in running speech. 屣 xǐ is to tread something flat the way you flatten the back of a sandal -- Wukong is promising to walk the mountain down.

Those little imps, hearing this, wheeled about in a fright, each fled back into the cave, shut the two leaves of the stone door, and came within to report: ‘Great King — disaster!’

扇 is shàn, the counting word for door-leaves, so 兩扇石門 is the two halves of the stone door. 禍事了, "disaster!", is the imps' standard cry of alarm throughout the novel -- you will hear it again every time Wukong turns up somewhere.