They were both, mostly in the early middle ages, heavily influenced by Chinese culture in the same way most western cultures were heavily influenced by Greece and Rome. Lot's of borrowed words, technologies, and styles (not to mention Classical Chinese holding a similar position in the east to that Greek and Latin once held in the west). Japan, in particular, still held onto a lot of its older traditions, and managed to alter and refine the Chinese traditions into something much different than had ever existed in China. The Japanese are good at that: now they've adopted a lot of Western stuff in the same way, while still remaining oh-so-distinctly Japanese).

The people who settled Japan and Korea came mostly from China a few thousand years ago and both of their cultures were heavily influenced by China up until early modern times.

No.

They both originate from the same culture, but that culture isn't Chinese.