Back when I was in South Korea, I had a few people tell me that Korean territory covered most of Mongolia and parts of northern China. I couldn't find anything to support this in any English written articles.
My Korean friend said it was called Ko Joseon.
This picture looks about right. However, I can't find Hwahn-gook in google.
I'll keep this brief and one of our Korea flairs can expand on the specific historical situation dealing with that period.
However I want to at least point out that Hwanguk (환국, 桓國) isn't/wasn't real. It's not actually accepted as a historical reality. It showed up in a book published in the late 1970s purporting to be a much earlier text. Neither the book itself nor the ideas of an ancestral Hwanguk are taken at all seriously. So at least for the image you've posted, it's not a thing. Also the Wikipedia user who created it also described it as "mythical", so that should settle that.
As for Gojoseon being that big, nah. While some accounts will draw the borders as larger than the Wikipedia page on Gojoseon that you linked, it certainly never included most of Mongolia. Parts of Northeast China is fair, because there was territory beyond the Yalu/Amok River, today's border between North Korea and the PRC. But not as big as people were apparently telling you when you were in South Korea.
Did Korea's claim extend pretty far from the current borders? Yes, but also keep in mind that that doesn't mean it was Korean to the same extent in those northern regions as it was outside of modern day Pyongyang. For example the Liaodong peninsula was controlled by Gojoseon for a time, but this was eventually lost. It was conquered territory, rather than some sort of ancestral homeland.
The "most of Mongolia" claim is honestly not one I've heard being taken seriously.
A couple of comments:
"Korean" didn't exist at the time so the people you spoke to were referring to the lands where the koreanic people's and people associated with koreanic culture dwelt.
The borders of ancient realms are difficult to track unless they border with a settled realm, meaning that the extent of western (mongolia) and northern (manchuria/siberia) cultural spread would be uncertain.
I've never come across any reference to "Hwan-gook" and any kingdom going that far back has very little evidence so I personally wouldn't take it seriously until something more concrete is found.
The natural boundary between Manchuria and Mongolia is a mountain range so it's unlikely that you'd have realms that controlled both sides of the range. Based on that I'd rule out the "most of mongolia" comment.
Manchuria though is well established as the homelands of koreanic kingdoms like Koguryo and Balhae and their semi-settled allies. It'd be safe to say that prior realms like GoJoseon similarly controlled most of manchuria.
Southwards, archaeological findings of violin daggers and dolmen across all southern manchuria and arching down to Shandong show a clear koreanic cultural influence across the areas across from the korean peninsula in present day china.
BTW, the map for gojoseon in wikipedia isn't based on archaelogical evidence because the greatest distribution of violin daggers is in manchuria, north of present-day beijing. That clearly indicates that those areas were centres of koreanic culture.
Here's a better estimate for koreanic cultural spread. It leaves out shandong peninsula, which might be because only dolmens are found there, but it gives a good minimum. The maximum spread north and west is hard to know as I mentioned previously.
https://byeongjupark.files.wordpress.com/2010/09/extent-of-gojoseon-11.jpg
So to summarise, well-established archaeological evidence shows koreanic influence of most of manchuria, the korean peninsula, and down to shandong peninsula.
If you go back further in history before GoJoseon there's Hongshan. PRC claims that Hongshan is the precursor of pre-historic chinese kingdoms but that's influenced by the PRC's political stance. More independent historians directly disagree with that claim, for example see:
https://youtu.be/3tMbp61lv9E?t=8m4s
In other words Hongshan culture is directly the opposite of the culture of the earliest "chinese" realm and therefore highly unlikely to be chinese. What is obvious though is that there was culture in manchuria that existed in parallel with proto-chinese culture.
Hong WonTack goes into further detail in his work on Manchurian history and the koreanic peoples. Here's a direct link to the early history section:
http://www.wontackhong.com/homepage1/data/1061.pdf
His work overall is quite well referenced and an interesting read overall.
Adding on to /u/keyilan...
Here is the wikipedia page in Hwanguk in Korean. My Korean is pretty weak, but even here, it points out that it's a myth.
Now, as far as actual size, from 698-926, there were two Korean kingdoms, Silla and Balhae. Silla controlled all of present day South Korea as well as Hwanghae and Gangwon, North Korea. Balhae controlled the rest of modern day North Korea, as well as most of Liaoning, Jilian, and Heilongjiang provinces in China, and Primorsky Krai in Russia. This puts modern day Vladivostok, Khabarohvsk, Harbin, Changchun, and Shenyang under Korean control (those cities sit roughly along the border of Balhae, ranging from on the border to 100 km inside of the border).
From 37BCE - 668, the kingdom of Goguryeo covered roughly the same as Balhae, MINUS the Russian territory, but including all of the Lioadong peninsula. Goguryeo engaged in some border wars with the Sui and Tang dynasties and traded fortresses/land along the Shuantaizi/Daliao rivers between 598-668.
Before then, there wasn't much else - the Korean kingdoms were smaller and not unified, and they were all within the same regions as what Goguryeo/Balhae would control later. In modern day China, there were the Buyeo, Xuantu, and Goguryeo. Modern North Korea had the Okjeo, Lelang, Daifang, Dongye. South Korea had the Mahan, Byeonghan, and Jinhan. If a lot of these names sound Chinese, it's because they are. They were military groups from China who controlled the areas and they were conquered, merged into, or fled from Goguryeo when they became the power in 37 BCE.
These locations have been confirmed through archeological digs that have found Korean dolmen (burial sites), wide bronze daggers, and narrow bronze daggers across the regions mentioned, as well as in the Fukuoka/Kitakyushu region of Japan.
So, to answer the questions:
I've mentioned this before when talking about China, and it's equally appropriate here. There was no "China" or "Korea" during the time periods we're talking about. There were clans who happened to take power at certain time periods in certain regions and went to war over other regions nearby, and their ancestors happen to live in the modern-day China and Korea. Goguryeo and Balhae didn't even control the entire peninsula. Goryeo controlled most of the peninsula, but it wasn't even until about 30 years into Joseon that the entire Korean peninsula was under Korean control.
A great book on this topic is Korean History in Maps: From Prehistory to the Twenty-First Century by Michael Shin. It places all of the kingdoms into context, showing their borders and cities, as well as modern city locations to give context. It also has detailed maps showing wars - army attack and retreat paths, ambush sites, etc. It uses a lot of Korean language primary sources, but it's all in English. Very accessible and clear.
