I am aware of Isabella Bird and her account of Korea, I have only read parts of her book. Very interesting account, her western interpretation of East Asia was very influential. Unfortunately she arrived in Korea in a time of absolute crisis. By 1898 Japan had nearly seized control of the country. Japan's wish of Korea breaking all ties to China and declaring itself independent had already occured in 1897. For Japan to annex Korea it needed Korea to break all historic ties to China. Nevertheless, Korea's ties to the Qing were tenous and fragile from their inception in 1644, I will come back to this point.
Since many foreigners only arrived in Korea at this time of crisis there were numerous misinterpretations of Korea that last even to this day. Such as Korea has "always" been the "shrimp between the two whales". Another common misconception is that Korea is a blend of Chinese and Japanese culture/civilization. Korea certainly has been heavily influenced from ancient China, less so from Japan. During the 1890's Korea was in a desperate situation, King Gojong sought assisance from Russia, China, and even the United States in its attempt to keep Japan at bay. Gojong sent Homer Herbert an American missionary and friend to the US on his behalf. (check the Homer Herbert connection) So at the end Gojong also asked the Qing to protect/assist them from the Japanese, as (Ming) China had during times of peril, such as the Imjin War. He tried to claim greater allegiance to (Qing) China than had ever existed to prevent Japan from taking control of the country.
Korea has pretty much always been independent of China, for much of the their history Korea was considered an enlightened neighbor whose King only had to acknowledge the Chinese Emperor as the "Son of Heaven" and offer tribute. Ming China and Joseon were particularly close, I would argue more so than any other Korean and Chinese dynasty. The Ming came to great aid to Joseon during the Imjin war when Joseon was threatened by Hideoyoshi's invasion. However, this relationship completely changed when the Ming Dynasty ended in the early/mid 17th century. To Joseon the Qing were barbarians who usurped the great civilization. Joseon viewed Qing (China) as uprooting the entire East Asian world. Joseon now only saw themselves as the bastion of that civilization, and looked only inward. Not long after the Ming collapsed (even with some support by Joseon troops) the Qing sought to re-establish the tributary system to an even greater extent than Ming China. This led to war as the Joseon dynasty would not accept Qing as a legitimate succesor, which led to a Qing army being sent to force Joseon to accept this relationship (which eventually did occur). For really the first time around 1644 Korea was now forced to accept a strong unwelcomed foreign influence, though Korea was still laregly independent. Check the history of Namhansanseong fortress, which is very central to this period. Qing laid seige to Nahansanseong for over a month in order to force the Korean King to submit.
For Korea, following the Imjin War and Manchu invasions began the long decline of the Joseon dynasty. Though there was still somewhat of a Joseon enlightenment later, the decline eventually led to the crisis during King Gojong's reign and Japanese colonization. Before these two major crisis in the early 17th century the old order of East Asia was very much well-established. China as the eldest brother, Korea as the middle brother, and Japan as the youngest brother. Before this time Korea's influence on Japan was far greather than any influence Japan held in Korea. In fact, Korea still often wrongly refered to the Japanese as barbarians and/or pirates. This came as a suprise to samurai invading Korea in the late 16th century who now saw themselves as the superior civilization.
I don't think a situation similar to Tibet would have occured in Korea. If you look at any Qing dynasty maps Tibet is a part of China, even if the Tibetans do not accept it. Qing maps nor Korean ones ever included Korea as part of China, this would be a transgression of the world order as they saw themselves. China throughout its history saw Korea as a ally, tributary nation, protectorate, a liability, a threat, etc but never really as part of the same nation. Particularly the Ming saw Korea as having a perhaps more established confucian system than even China, with Joseon scholars and aristocracy heavily focused on Confucian principles in governance.
Moving on from the history and approaching your question. "So - how and when did the northern/North Eastern frontier between China and Korea get established?"
Well during the Joseon dynasty the Yalu river was the well-established border between China. For the Northeast frontier the Tuman River was the border. From what I looked up there were only minor changes of the original Joseon border from the Empire of Korea period to the end of the Korean War. Most of these changes, like you mentioned were attempted by Japan.
When you finish reading Isabela Bird's account, I would reccomend Homer Hulbert's accounts of Korea. I think more than any other westerner of the time he understood Korea, and lacked a pronounced bias seen in other Western accounts. He spent over 20 years in Korea, travelled the entire country, wrote and spoke Korean fluently and knew King Gojong on a personal level. He even wrote a newspaper in Korean! He was an advocate of Korea, and spoke out against Japan about the illegal nature of their actions against Korea. It is almost like Homer Hulbert is from another time, it does not seem like he is from the 19th and early 20th century. His understanding of East Asia is so much more advanced then most Westerners who viewed East Asia as "exotic and backward" and were often perplexed by what they observed in Korea and China.