•why were nuclear weapons important during the cold war?

Off the top of my head:

The US devised the Manhattan project during ww2 and afterwards to develop nuclear weapons. The kept this secret from USSR even though they were allied. This contributed to Stalin’s paranoia and increased tensions between the two countries.

You also mention how Truman used this ‘atomic-monopoly’ to give him confidence and make him think that he could dictate decisions during the Potsdam conference and other meetings which heightened tensions with USSR.

Nuclear weapons also played a large role during the Cuban Missile Crisis, the point where nuclear war was at it’s closest to becoming reality and destroying the world.

By that point the USSR had also developed atomic bombs. The fact both superpowers had nuclear weapons meant they had to be sensitive in the way they handled each other and you could link this with Cuba and argue that it was the only reason the Cold War didn’t turn into full-scale, physical war.

There’s other things you could say beyond these points as well.