Primorksy Krai is not part of Korea and it never has been.
Long-term Korean control of the area is unlikely at best.
Depends on what you mean by "Korea."
Gojoseon and Goguryeo together controlled a significant amount of Manchuria for about 700-1000 years, and about half for about 300-500. Succeeding dynasties, namely Silla, Goryeo (which was originally another name for Goguryeo), and Joseon, adopted the previous dynasties' cultural aspects. Buyeo was also located in eastern Manchuria, but was eventually absorbed by Goguryeo. Neither Gojoseon nor Goguryeo directly controlled Primorsky Krai, which was occupied by Mohe tribes, but the people living in the area often allied with Goguryeo, and were probably culturally influenced by the two dynasties. Goguryeo itself was ethnically diverse, so they could have incorporated their neighbors more thoroughly if they wanted to. Balhae, which was founded 30 years after Goguryeo fell, probably consisted of more Mohe than other ethnicities, but culturally and linguistically copied Goguryeo traditions, so it's hard to say what kind of state it really was.
It's not that hard to think of a scenario in which Primorsky Krai could be absorbed into a Korean state: have Goguryeo politically unify the peninsula by 400-450. This would bring the total population to about 10-15 million, and because the Mohe probably did not number more than 5 million at the time, it's likely that they would be gradually absorbed. The other possibility is Goryeo asking the Balhae refugees for historical records and maps, then informing the Liao of their possible territorial claims. This is less likely because Goryeo probably did not have the military nor demographic capability to establish control over a significant portion of Manchuria, and the Liao would definitely not want to hand over recently conquered territory so easily. However, given certain conditions, including diplomatic negotiations, which were frequent in OTL, it is slightly possible.
Another scenario would involve Gwanghaegun retaining power until his death, which could lead to a different Ming-Qing conflict. Gwanghaegun could have reorganized the military efficiently, and come to the Ming's aid by attacking Manchu homelands, which could have led to the latter's collapse, or a rump state north of the Great Wall. In either case, there are two states immediately south of Siberia, which means that additional butterflies could lead to the region's exposure to Manchu and Korean influences before Russia established permanent settlements in the area. The result of the conflicts would ultimately be based on which state would be able to populate the region more thoroughly, but with the right conditions, Russia might end up having limited control, although it might be hard for the natives to establish a state with a sparse population.