I disagree. There was only a single invention of gunpowder and every subsequent development and usage can be traced to that. The Mongols and Indians got it from the Chinese. The Arabs got it either from the Mongols or the Indians. The Europeans got it from the Arabs. There was no independent invention.
Essentially, gunpowder was an outgrowth of Chinese medicine. Chinese medicine was obsessed with attributing and cataloguing medicinal properties for literally hundreds of substances, some of which (dragons teeth/fossils) were entirely spurious, or (rhino horns for potency) nonsensical or simply (mercury) misguided. The attributions were often somewhat arbitrary. Some of it worked, or at least enough of it seemed to work to sustain an intellectual superstructure that was dramatically wrong.
Having catalogued medicinal properties up the Wazoo, there came the Taoist notion that perhaps the proper combinations could lead to wellness and indeed, immortality.
Gunpowder was discovered as part of the quest for an elixir of eternal life. Early recipes of gunpowder included honey.
But what it came down to was that gunpowder was originally mixed pursuant to cockeyed and wrong theories of medicine, from a catalogue of literally hundreds of substances.
So to mix the four key ingredients together (honey, nitrates, charcoal and sulphur), when in fact there were literally hundreds of ingredients to select among, was basically a relatively improbable fluke. Mixing them together in the correct proportions (or close enough to get a significant result) was an even bigger fluke. Combusting them, another fluke. And publicizing the result, yet another.
In short, it's incredibly easy for the Chinese to miss out on gunpowder altogether.
Of course, the corollary, is that its just as possible for any early civilization which had some grasp of the critical constituents to mix them up and stumble onto gunpowder. For all we know, the Sumerians or the Inca might have been able to do it.
The second part of the question is tricky, since the absence of chinese gunpowder butterflies history as we know it.
I would say that allowing history to proceed more or less along similar lines, the medieval alchemists might have had a similar shot at an inadvertent fluke gunpowder.
Otherwise, maybe the 18th or early 19th century, where there seems to have been some serious efforts at understanding and analyzing the world, and a tradition of directed experimentation, as a result of the enlightenment.
Keep in mind that the discovery of gunpowder during this period would not result automatically in firearms. Those took several hundred years to achieve in OTL. The path might be considerably shorter, but there'd still be a huge learning curve.