During the Cold war, was there a danger for "Third World War" ?

Yes especially twice. Once was the Cuban Missile Crisis in October 1962. The crisis ranks with the Berlin Blockade as one of the major confrontations of the Cold War and is generally regarded as the moment in which the Cold War came closest to turning into a nuclear conflict. It also marks the first documented instance of the threat of Mutually Assured Destruction (MAD) being discussed as a determining factor in a major international arms agreement.

The United States considered attacking Cuba via air and sea, and settled on a military "quarantine" of Cuba.

The other was of course the Berlin Crisis On June 4, 1961, Premier Khrushchev caused the crisis he threated to sign a separate peace treaty with East Germany, which he said would end existing four-power agreements guaranteeing American, British, and French access rights to West Berlin. with a deadline of December 31, 1961. The three powers replied that no unilateral treaty could abrogate their responsibilities and rights in West Berlin, including the right of unobstructed access to the city.

The crisis got to the point of US and Soviet tanks facing off each other at Checkpoint Charlie. With units on both side being moved forward.

No. The threats of worldwide communism was greatly exaggerated. Russian communism was unsustainable the larger it grew. As for the threat of a nuclear arms build up - both the U.S. and U.S.S.R. understood that these weapons were a deterrent and never attended as offensive weapons.

yes , but the major threat was nuclear war