Your question kinda mixes a few different things. We need to make two very clear distinctions before we go any further, and that's between writing systems which may have been borrowed a bunch, and spoken languages which often do stick to one writing system at a given range of dates but aren't necessarily related to the languages that use the same writing system (e.g. Vietnamese using the same writing system as English)

Let's start on the writing system.

Everything you've said about the writing of Chinese applies to Japanese, Korean and Vietnamese as well, except:

• Japanese eventually developed a syllabic writing system, two actually, being katakana and hiragana. Katakana looks like カタカナ and hiragana looks like ひらがな, which we can simplify as not curvy and curvy. However both of these are ultimately derived from Chinese characters. Still, they're something different, and they're something created for Japanese.

• Vietnamese switched to a roman script. It looks like this, tiếng Việt, with a bunch of diacritics. This was a system devised by French missionaries which replaced the old system, being Chinese characters.

• Korean now has a system called hangeul, which looks like 한글 자모. It has a questionable origin, mostly because it's been turned into a tool used by nationalists, but it likely is at least in part derived from an old Buddhist script called ʼPhags-pa, and has clear influence from Chinese characters as well. It was developed centuries ago but only really became widely used more recently, outside of a few specific things like pulp fiction novels mostly read by women. Even as recently as the 90s, Chinese characters were used a lot, at least in certain contexts. Some are still used today, albeit mostly as abbreviations. It is credited to King Sejong, although his role was more in sponsoring it than creating it himself.

In each case, prior to the change, they were generally writing in Classical Chinese, using Chinese characters, and also incorporating a number of their own character forms. In Vietnamese these were called Chữ Nôm 𡨸喃. In Korean, a number of different systems existed based on Chinese characters. That's a whole separate thread on its own. Suffice to say, people came up with character forms that weren't actually Chinese but existed in the same system in order to better reflect their own languages.

We know heaps about how these developed, when they were used, etc. We have written records for all of that.

The second part of the question is on the spoken languages. This is an entirely different subject.

Japanese belongs to a family called Japonic, which includes other languages of the Japanese islands. Korean belongs to Koreanic, which also includes Jeju and a bunch of extinct languages. Some people argue that they are related families. There is, at this point in time on a Tuesday in July in 2019, no actual evidence that they are, at least in the form of scientific evidence that would be accepted by historical linguists. Specifically, there are no historical sound correspondences which are required for them to be demonstrably related. They might have been at some point but the evidence as gone and we'll probably never be able to show it.

Vietnamese belongs to the language family called Austroasiatic or Mon-Khmer, which includes also the official language of Cambodia, called Khmer, as as well as Khasi spoken in India and Wa spoken in Myanmar. It's a wide-ranging family, geographically. It is not related to Japonic or Koreanic or Sinitic (Chinese). However they used Chinese characters for a long time.

Anyway, we have a really good idea for all three of these how they evolved. We have great work done by excellent linguists on the development of all three. I won't get into the details just now because I'm not sure what details you'd be interested in, but we know about as much as those three as we do about Sinitic.

Obviously we can't really address origins, because that's actually the ultimate question in historical linguistics: Did language happen once and all are related, or did it come up independently in different places at different time like writing did? We don't know, probably can't know for sure in our lifetime (but hey who knows, maybe that'll change). But we can at least place Koreanic, Japonic and Austroasiatic somewhere on the timeline. There will be some debate about the exact timing, because again we can't actually know for sure in many cases, but we can have a pretty good idea.

If you have specific questions beyond that stuff I'm more than happy to answer. I think otherwise I'd best stop here or I could be typing this comment up all night, just for how broad and open-ended the question is.

Hope that helped. Let me know if you have follow-ups.