Speaking of which, did the Mongols invade Taiwan back in the day?
I don’t know about Taiwan, but the Mongols did try to invade Japan. Their fleet was destroyed by a storm — the Japanese called it a “Divine Wind” or “Kamikaze”.
Nope. Chinese dynasties paid little attention to the island before the Manchu.
They almost certainly could have, but it was still very primitive and not much point to doing so. Nothing to steal.
I don’t think so. But by defeating the Sung [the last boy emperor jumped overboard during a sea battle, they may have gained suzerainty over the island, if it was part of China at the time.