I'm especially curious about their interactions and borders with client states, like Vietnam or Korean kingdoms.

Do you mean the Han here, with their relationship with the peripheric states of Chaoxian and Nanyue? Because in the title you mentioned the Zhou and the Qin, and:

The Zhou state was "feudal," for lack of a better term, with little in the way of direct centralized control for most of its realm. So in a sense the Zhou state was largely composed of "client states." However, Korea and Vietnam were never part of the Zhou world; indeed the Zhou did not include regions that are now integral to China, such as Sichuan (see Ancient Sichuan and the Unification of China for a bit on ancient Sichuan). Indeed actual state formation in Korea postdates the de facto loss of Zhou power in the 8th century BC. Vietnam I know less about, except that it was not Zhou either. Actually, the pre-Qin state closest to Vietnam was Chu, which was the first to assert independence from Qin and was generally not as "Chinese" as the Zhou center (Defining Chu is pretty good about the Chu).

For the Qin, they set up commanderies in parts of Vietnam as did the Han, but even under the Han the area remained only loosely connected to the imperial center in North China. The Qin never invaded the Korean states and we know little from the histories about how the rise of Qin affected Korea.

ETA: On mobile, sorry if this is a bit incoherent.