The "possibly exaggerated" is more of a certainty than anything else about a lot of the Warring States period, is all I'd say. I mean there are accounts that claim hundreds of thousands of soldiers on each side but estimates say that those are unfeasible even for the total number of men in the field.

Sure, that could work too. It's very fluid after all, no exact date. Same as the slow decline of the Western Roman Empire is quite fluid

As with all things it depends where and when you are.

"The Tudor era" only really applies to England.

Some scholars would argue broadly the medieval period ends in 1453 because that's when the 100 Years' War ends, and when the Ottomans finally topple Constantinople. The world after that was a quite different place. Warfare also had changed drastically by then.

Some argue for 1492 as a good date.

Others say 1521 and the start of the Protestant Reformation is the end of the medieval period.

Personally I can see the argument for each but I like 1453 as the date.

From that range I mean... there's more years there that are post-medieval than actually medieval...

Some have touched on school in the ancient world so, I'll go to the Medieval West, starting in around the 8th/9th century and beyond for a bit...

School was most often run by the church or church adjacent institutions [like a monastery or, a group of monks/priests in the employ of a King, say] and as such would require study or religious texts, as well as the "seven liberal arts" - the Trivium and the Quadrivium, if you've heard of those.

The trivium was grammar, logic, and rhetoric. In studying this, most fledgling monks or lay people would probably be reading a lot of Aristotle and perhaps a smattering of Cicero or other Latin writers. Meanwhile, the quadrivium included arithmetic, astronomy, geometry, and music. Here again, classical authors would have been read and preserved, along with some late antique and medieval writers as well.

Starting in around the 11th/12th centuries, monastic schools started to develop into larger institutions, which eventually gave rise to the earliest universities [oversimplifying some]. The seven liberal arts still made up the core of one's education which could lead to a bachelor's degree. A Master's degree might focus more on the quadrivium, or an area within that, and after attaining an MA, one could study theology, medicine, or law at higher levels.

So as you can sort of see - you wouldn't see people studying history as its own field. History would come up when studying theology, when reading for rhetoric and grammar, and in other ways tangentially, but it was not a focus.