Our understanding of the past is reliant on things others wrote down, yes. What are the things written down, and by whom? That varies dramatically with the event, time period, context, etc. Sometimes we have remarkably good written sources, even for things very long ago: multiple, independently-derived accounts of an event, that allow us to feel rather confident that some version of the truth is reflected in said records. Sometimes we have just one source and it is not obviously very reliable: one has to tread very carefully in such a case, because we all know that an individual person can be very untrustworthy, or just wrong, for a million reasons.

The length of time in the past is not as important as the question of whether there is adequate documentation that survives. In the case of China, you are talking about a nation that not only had writing for a very long time, but you are talking about a state that was highly bureaucratized relatively early in human history. So they had things like a census, official records, tax assessments, official histories, and so on, much earlier than some other parts of the world did. The Chinese were exceptionally interested in earthquakes and developed the careful study of them far earlier than anyone else (and the first seismographs, etc.).

I don't know the specific document sources for those particular topics. But you can find them relatively easily: find yourself a scholarly book on the subject and look at the footnotes/endnotes. They will tell you exactly where this kind of information comes from, and will frequently discuss (in far more careful terms than a tertiary source like Wikipedia) the limitations of what is known and the limitations of said sources.

Casualties on the scale of millions are frequently hard to know with reliability even in modern wars (consider how much uncertainty there is about civilian casualties from the Iraq War), so there are typically large error bars around such numbers, and such things are frequently debated by scholars. One can roughly say, "if this region/city/state had X million people before the war, and Y million people after, then it must have suffered a great much from it," but again even in the modern world this is tricky (because populations move in wartime, for example). This doesn't mean that such numbers are nonsense, but it does mean that they are frequently contested.

Again, if you are very interested in understanding the evidentiary nature of a given claim, look to the footnotes. Wikipedia will typically only cite scholarly works ("secondary sources") because that is its policy to avoid cranks ("no original research"). But if you find the scholarly works they will eventually trace their claims back to a primary source of some sort.