About 5,500 years ago, the ancestor's of Taiwan's current aboriginal population migrated to Taiwan. From that time on there was considerable trade and communication between Taiwan and the mainland, between speakers of Proto-Sinotibetan (and varieties developed from that) and Austronesian. Some of these people on Taiwan left and returned to mainland Asia, where they had even greater contact with the Sinotibetans (ancestors of today's Chinese). Trade betwwen Taiwan and mainland Asia has been ongoing since before recorded history.

Throughout China's history (which purely for the sake of argument we'll say is 5000 years), "China" has known Taiwan was there. It wasn't ever in need of being discovered since it's peopling predates written records and exchanges between it and the mainland were never wholly cut off for it to be re-discovered.

So no, they didn't discover it.

At least as far as written history, people were always there. There's rome evidence of a yet earlier population of non-Austronesian peoples but I can't really speak to that in detail.

We should also be clear on what we mean by "China" in your question. What's called China today is a different nation than what was called China in 1920 and a still different nation than what was called China in 1890, even though come f the same people lived through all three.

As for acquiring it, they didn't, exactly. The Qing annexed Taiwan in 1683 and had nominal rule. There was a lot of unrest in Fujian and surrounding areas resisting the Qing rule, and annexing Taiwan was in part an effort to fight piracy based out of the island. They didn't do much with it though. Then in in 1895 the Qing gave it to Japan, and in 1945 Japan was made to give it to Republican China (the current Taiwanese government being a continuation of that).

So China in the modern sense of the PRC has never had Taiwan. Only during the Qing has it been a "Chinese" administrative division. The reason for the quotes there is that it's important to remember that the notion of what is "China" is not a black and white issue and the Qing rulers being not Chinese was at the time a significant issue, all the way up to the fall of the empire.