What did Mongolia do during the cold war?

All of Mongolia was part of China until 1911, when Outer Mongolia (the northern part) became independent. In 1921, it became a communist satellite of Soviet Union along with 2 separate Mongolian states, the Tuvan Republic and the Buryat Republic, all of which were only recognized by the Russians. In the 1930s, Tuva and Buryat were annexed into the Soviet Union, but Mongolia was left out as to not offend Chinese. It became a member of the UN as a compromise with the Chinese. Mongolia was a communist country until 1991, with almost all of its trade with the Soviet Union, several hundred thousand Russian workers running most of its industry including a large copper mine in Erdenet, the Trans-Mongolian railroad, and its cashmere industry, as well as 100,000 Soviet soldiers guarding against the Chinese. The copper mine and railroad are still 50% owned by Russia.

Mongolians didn't have any significant role in the cold war outside clashes between China and the Soviet Union when Stalin and Mao fell out.

Mongolia was a satellite state of the Soviet Union from the 1920s until the end of the Cold War. During the Cold War, it was an integral part of the Soviet bloc.

The modern nation of Mongolia had been the state of Outer Mongolia in the old Qing Empire that ruled China prior to 1911. It regained its independence after the overthrow of the Qing, although to this day, the government of the Republic of China (Taiwan) considers it to be rightfully a part of China. By aligning with the Soviets, Mongolia assured itself that it would never be taken over by China again. The country hosted Red Army troops through the entire period, who even defended Mongolia from a Japanese attack at the Battle of Khalkin Gol in 1939.

Inner Mongolia was annexed by China. Outer Mongolia became a Soviet satellite and was largely exploited to assist the Soviet economy. Its main export was a low grade brown coal and the products of traditional herding, mostly of sheep. It was Communist in name but in effect a colony run by a dictatorship. Apart from the USSR and China no other nations bordered Outer Mongolia so it was not involved in the Cold War. Any disputes between the two Communist Powers were muted.

Yes, it was a puppet state of the Soviet Union. Its political leadership aligned itself with the Soviet Union, until the collapse of the Soviet Union in the early 1990's.

It was not neutral in so far as it had a military pact with the Soviet Union, assuring that any country that became aggressive towards it, would be viewed as aggression against the Soviet Union.