So i saved this just in case nobody answered it and it seems like no one has, so i will try and field the bat based on what i've read from korean books.

In terms of Jin, I believe they were seen as barbarians. Balhae, which ruled over the area before Jin dynasty was seen as successor to Goguryo just as Goryo saw itself as a successor to Goguryo as well. If im not mistaken, Goryo had wanted to absorb Balhae before it fell to Jurchens(Manchurians) but could not save it in time.

As for Liao, Goryo had several conflicts with them and they are seen as tribal in nature(and barbarians) and occupying what was their territory as Balhae. Some of the more general books focus on the Goryo-Khitan wars and eventual truce between the two. For Koreans, any army outside of ethnic Chinese and Koreans themselves are seen as Barbarian as per sinocentric order(Although this could be due to the historiography of the Joseon court considering about half of Korean history is based on fighting against the Chinese)

Korean view on Mongol invasion is essentially the same as any other barbarian invasion(like the Manchurian invasions). without a doubt at the early stages of the Mongol invasion, they were seen as another barbarian horde, which would explain the killing of Mongolian envoys. Im inclined to believe that "racial" factors went in to motivating the Koreans to not give up on resisting the Mongol rule which forced the Mongolians to invade the Korean peninsula four times. After their last defeat, which exhausted the Koreans, they essentially agreed to be subjugated in return for semi autonomy. I've also read that Mongolians and Koreans saw each other as distant kins, which sort of explains somewhat generous treatment by the Mongolians to the Koreans.

Lot of what i wrote is based on what i've read in Korea. Be aware that racism is still rampant in Korea, so you should take what i wrote with a large bag of salt.

Relations between Koryŏ and the Liao were utterly horrible, with the latter invading the former several times. Koryŏ did not like the Khitans, unsurprisingly. However, this should not be taken to mean that cultural exchange between the Liao and Koryŏ ended. I've read Liao influences influenced Koryŏ in religion and architecture, and that's probably not the limit. There were also border disputes between the two states about the land along the Yalu River. Even though Koryŏ survived the Liao invasions, it became a Liao tributary, though the effects of tributary status should not be overstated.

However, relations with the Jin were much better. There were no wars between the Jin and Koryŏ. One reason is because the Jurchen Jin felt a sort of affinity with the Korean Peninsula. The founder of the Jin royal family, Hanpu (函普) is recorded as joining the Jurchens after coming from Koryŏ. The other reason is because the Koryŏ state was much less interested in conflict, because the wars with the Khitan were devastating and the conflicts with the pre-Jin Jurchen were not so fun either. Koryŏ thus accepted a tributary relationship less reluctantly, but not everyone was happy about this. The monk Myoch'ŏng (妙淸) really didn't like this situation. He wanted Koryŏ to declare equality with the Jin and Song. The Koryŏ court decided that was a bad idea (and crushed Myoch'ŏng when he revolted over this issue).

The Mongols allied with Koryŏ against the Khitan, and I've never read anything that said this alliance extended to fighting the Jurchen. The relations between Yuan and Koryŏ are complicated and deserve a separate question.