During the Cold War, in everyday conversation, was West Germany referred to as "Germany" like South Korea is currently often referred to as "Korea" and the People's Republic of China is currently often referred to as "China"?
My experience only covers international media at the time (to some extent) and colloquial speech in German, within West Germany. Nevertheless, I think this may help:
As I remember it, it was not handled very consistently. I think the typical approach was to call it West Germany on the first mention and become very inconsistent afterwards. If the context was clear and the medium relatively informal, it may have been just Germany from the beginning. Some people were more consistent about the careful distinction for political reasons -- typically as a reminder that there was a political problem there. (In Western Europe people tended to almost forget about this.)
I think a somewhat similar situation (obviously without the political dimension) is that of the US vs. America. It's inconvenient to talk about the United States all the time, so America is used as a synonym even though strictly speaking that's a much larger area. The US is just the most important part of America for most English speakers. But West Germany wasn't quite as dominant, and also East Germany also laid claim to the same name. So use of "Germany" for West Germany tended to be a bit more limited than use of "America" for the US.
By the way, it's not true that people had virtually no contact with East Germany. They did have some significant exports, and in sports they were in a close competition with West Germany (in terms of olympic gold medals and such).
Yes. Press archives are a good way to confirm this, e.g.:
The greatest moment in the history of English football came at 5.15 this afternoon when Geoff Hurst shot the magnificent goal that made certain of the World Cup. It was Hurst's third goal, England's fourth, and, coming as it did in the final seconds of extra time, it shattered the last remnants of German resistance.
Germany had equalized with the last kick in the regular 90 minutes, and they had gone within inches of repeating the blow in extra time when Seeler lunged in on a headed pass by Held. But Moore took the ball coolly out of defence and lifted it upfield to Hurst 10 yards inside the German half. The referee was already looking at his watch and three England supporters had prematurely invaded the pitch as Hurst took the ball on his chest.
At first he seemed inclined to dawdle out time. Then abruptly he sprinted through on the inside-left position with a German defender pressing him. As Tilkowski prepared to move out, Hurst swung his left foot and drove the ball breathtakingly into the top of the net.
From the Vault: Hurst's hat-trick wins the World Cup, The Guardian, originally published 1966.