Ancient/Medieval/Modern is a set that gets applied surprisingly frequently to India and China. For example, you might find "Ancient India" being defined up to the dissolution of the Gupta, and "Medieval China" being from the fall of the Tang to the Song. It isn't meant as a rigorous term, more as a quick shorthand.

As far as I know, everyone understands and uses the ideas of "prehistoric" and "neolithic"; most use "copper", "bronze", and "iron" ages because they did in fact go through that order as they developed agriculture and a smelting industry.

But no, India, China, Japan, &c. do not generally divide their own histories according to the political and religious situation of Europe. (The Americas and Australians usually do, but that's because their societies are largely made up of European descendants and they treat European history instead of the largely unrecorded histories of their native peoples.)

Instead, you see a mix of systems. The most common is based on the major dynasties of rulers—or, in Japan, the clans of military leaders who controlled the emperor—so, e.g. Chinese histories might talk about how the Roman Empire occurred under the Han or that Newton worked during the early Qing. The Han was a high point, sure, but so were the Tang and Song, so you never see them cluster together all the dynasties between the Three Kingdoms and the Ming as the Bad Times.

Modern China is formally Communist, so you also see its history fitted into Marx's Western scheme... but on a completely different time scale. It's common in Chinese to use the character 古 (lit., "ancient")—the same word they use to talk about Ancient Rome or Ancient Greece—to talk about the part of the 20th century when the Qing still ruled the country. Meanwhile, when they discuss "feudalism", they use the Chinese term 封建 and stretch it back to the Zhou dynasty two millennia before the High Middle Ages.

Not necessarily--the countries/cultures that do have same or similar historical periods as the West tend to have already had heavy western influences in their academic institutions (ie: former colonies or imperial holdings of western powers). Of those that do differ, the time spans themselves might not vary too much while the names of the periods and the interpretations of those periods would.

Here is a book I would recommend on the subject. Since first reading it, I have used this as a starting point many times while doing secondary research. It's a bit dry and jargon-y though; a good skimming would definitely do it enough justice.

Sorry if this doesn't help too much, but it would be a good place to start.

ED: a letter