Structurally, Japanese and Korean is almost identical. The morphology system are similar, they have the same word order, are both tonal in a similar way, and there are even parallels that can be drawn in terms of their phonology; some words are even phonemically identical. However, their core inventory of phonemes is quite different: there are sounds in Korean that just do not occur in Japanese. Like Japanese, the categorization of Korean is debated; Wikipedia is helpful here if you want to know more.
Now, Mandarin Chinese is entirely different. The tonal system is different, it lacks an overt inflection system, word order is different, and the phoneme inventory is, again, different. While there is vocabulary sharing going on, it's hard to see any technical relationships between them.
I've heard a number of different stories on how the Korean peninsula and Japanese archipelago were settled by the Chinese, from the Chinese, but it just doesn't seem plausible considering the linguistics of the regions. What history I did study in Japan simply never mentioned it. And, to be honest, that is the extent of my historical knowledge of the languages.