Since the area around Berlin was Soviet(it was, right?), how would West Berlin even trade with anyone outside? Somekind of deal to get permission through Soviet territory? I've no idea how that worked.
Also, with same relation to the territory; how would it even be possible to take a part of Berlin but not the area up to West-Berlin?
Berlin was divided after the war and, as the capital of Germany, was sectioned off betwixt the Allied powers. The reasons for this are complex and beyond the scope of this response. Suffice to say that it was a polictical decision, not one borne of the particular battle lines when the shooting stopped.
As to the trade with and supply of Berlin, it certainly did require the permission of the Soviets/East Germans, at least later on. The Berlin Airlift, a US lead round-the-clock air-freight operation in West Berlin in 1948, was in response to a Soviet blockade of Berlin which was sparked by the introduction of the Deutsche Mark by the US and its allies. (The Soviets wanted to use the old currency and then the Ostmark and presented the whole issue as an economic one.) The Berlin Airlift was an operation to supply the city by air and it was successful, but only just. West Berlin could survive without the passage of supplies on East German road and rail, but not easily and not for long.