I don't know very much about the Imjin War yet, though I am hoping to look more into it once I have time and have finished some other books I'm reading. However, from what little I do know about it, one thing confuses me — what on Earth was the goal?
From what I understand, Japan was aiming at conquering China by first going through Korea, but got defeated primarily by Admiral Yi and his turtle ships. (this is a really limited understanding of it all, I know). My question is, why did they want to conquer China (and/or Korea? I still don't understand if they were merely wanting to pass through there or something else)? Would it have provided some strategic advantage? Or was it just because they wanted more territory, and if so, why did they want it? Also, how did they imagine that they, located on a small island, could conquer China, which was obviously much larger and thus likely had a larger military (or at least a larger population to draw soldiers from)?
I know that this is a lot of questions merged into one, but generally I'd like to know why Japan was trying to attack China (and Korea??), what the ultimate aim on their side of the Imjin war was, and why they thought that fighting a war with China was even feasible.
Yes, Hideyoshi did aim at conquering China. You can read about that and why he might have thought he could do it here.
It's clear that by 1593, after China entered the war and early in the first round of peace negotiations, Hideyoshi had realized it's not feasible, at least in the moment, and he changed his expectations to just the southern half of Korea. I talked a bit about the first round of peace negotiations here.
Hideyoshi's war plans in 1597 were to hit from their hold on the southeastern coast into Chungcheong and Jeolla provinces (southwestern quarter of the peninsula) and then construct fortifications and hold them. The list of locations to hold issued on the twenty-first day of the eight month don't survive, but the order gave the commanders free reign to choose the site should they have advanced further than ordered. The final deployment agreed upon on the sixteenth day of the ninth month at Jeongeup (coincidentally on the same day as the Battle of Myeongnyang, but word of the Japanese defeat most likely didn't reach the meeting and effect the decision due to the distances) was to spread the Japanese fortifications in Jeolla and Gyeongsang provinces, the two along the southern coast. Since Hideyoshi did not tell them to redeploy, it seems they either followed the original order or he was fine with where they ended up. It therefore seems he only wanted to hold on to the southernmost two provinces, at least in the short term to see how the war would play out. Or perhaps by this point Hideyoshi, now on his deathbed, was too busy trying to set up the political order after his death to pay attention to details at the front.