The Cold War was not a real war and people who claim so should be lucky they're wrong.
Yes. If it was a real war, we wouldn't be able to claim that it was a real war.
It depends on how you look at it. The Cold War was actively fought by proxies for the most part. But, that doesn't make it less of a war. Between the Chinese Civil War, the Korean War, the Vietnam/Indochinese War, the Yom Kippur War, the various Latin American wars, the Soviet invasion of Afghanistan, there was good deal of war going on around the world between two major alliances.