What exact countries were also involved in the Cold War other than the United States and Soviet Union?
It is not possible to list every country that was involved in the Cold War because there was no official Cold War that countries could be involved in or not involved in. Countries did not declare war on one another in this conflict so there is no official list of countries that were involved.
In a very real sense, every country in the world was involved in the Cold War. Every country was under some amount of pressure to side with the US or with the USSR. Therefore, we had countries like the Philippines being involved in the Cold War in the sense that they chose to side with the US and fought against a communist insurgency within their country. We also had countries like Australia that were involved in the Cold War in the sense of being on the side of the US and helping the US in conflicts such as the Vietnam War.
Some countries, like Egypt, explicitly tried to remove themselves from the Cold War, creating the nonaligned movement. But even countries like this ended up being involved in some ways in the Cold War. When Egypt and Israel went to war briefly in the 1956 Suez Crisis, the superpowers got involved with each side working to rein in one of the combatants. In this way, even countries that tried to opt out of the Cold War were often dragged in.
So, the best answer to this is to say that every country in the world was in some way involved in the Cold War.