Yes other countries participated as well. You may have heard of the NATO nations. They were on out side of the conflict and it was a strategy to stop, threaten, and contain the USSR.

We shared information, and in regards to your question, we also shared nukes and nuclear technologies which is why Britain, France, and other nations have stockpiles like we do.

The USSR supported nearly all countries that were or wanted to be communist such as Korea, Cuba, Vietnam (see a pattern). These countries we often fought in under a containment strategy. Basically, don't allow communism to spread any further than it has already.

This created a series of wars that did not happen between the nuclear powered countries but instead their smaller counterparts.

The one communist country that the USSR didnt support was Maos china. China had a different philosophy on how communism was to operate and neither side cared much for the others brand. As a matter of fact the USSR was fighting a Cold War on two different fronts, one with the US/NATO, the other with china which is an interesting study in itself.

Another thing you might be familiar with is the term first world country. That has nothing to do with wealth or development initially. Instead it was to define which sides countries fell on during the Cold War.

The United States and it's NATO allies were first world

The USSR and it's various allies were second

And the third world countries fell into the category that were on neither side. Basically if you didn't fall into one of the camps it's probably because you weren't important enough to garner attention (African nations etc) or you were neutral.

I hope that helps you out.