So I did laugh a little while reading this, because this must be fate playing around again. The Mongol Empire began after Genghis Khan's descendants finally conquered the remains of the southern Song Dynasty, and that happens to be the same Dynasty my fictional story is mirrored with! Whats even more hilarious is the fact that I was planning to incorporate some events from the Mongol invasion of Song in my story!
Regarding your questions, I am actually Chinese, though I only lived there for about 3 years before I moved back to the US 2 years ago. I was shocked to see how little I knew of my own culture, so I talked a lot with my high school mandarin teacher and did a LOT of research. Even now, after coming up with the idea of TSPBT and writing it, I still have just whole research days where I gather information on all sorts of things!
If you want to learn more about dynasties, I would definitely ask some native Chinese people if you know any, and if not, certain historical websites like Brittanica have helped me. Or, I could tell you what I know, which is mostly about the Song Dynasty and mythology! But one thing is for certain - if you want to be sure you have accurate information while researching, read several different sites about the same topic and pick out what they have in common. Brittanica is usually pretty good but sometimes it leaves out certain events and has different versions of things.
Also, don't try to absorb it all at once! There are 13 major Dynasties of China and countless smaller ones, so just have fun and take it at your own pace! (^∀゚)ъ