What are some major military successes achieved by the former Soviet Union against the Western World that shook the US and its Cold War allies?
There were none. Likewise, the U.S. never scored any military victories over the Soviet Union.
The Cold War was a war by proxy. One superpower was a combatant in Afghanistan, Korea, and Vietnam, but the other did not send troops to the other side — partly out of fear of escalation into global conflict. Rather, they provided support to their allied local factions, as both did in the numerous other conflicts of the Cold War.
That is not to say that other military conflicts did not shake the Western powers. The single most significant development was the defeat of the Nationalists in the Chinese Civil War and the declaration of the People's Republic of China in 1949. The shock and the ensuing bitter debate over who had "lost" China colored American policy for a generation.
The answer to your question depends heavily on how you define "Military". As Choster implies, the Cold War was never "hot" by definition, so that excludes any direct military confrontation, and thus no soviet military "successes" against the West if we define a success as such.
But given that you are asking the question, I assume you are not defining it in that narrow way and IMHO, it is absurd to argue a military cannot achieve a success against its opponents except in active combat since that excludes any peacetime activities of that military which might nonetheless have a huge effect in an economic, technical or strategic sense. Military funding is not provided just to fight wars and train for future wars after all.
So, to your question, there are two areas of achievement. I'm sure we could find more.
One, the Soviet space design bureaus were in large part driven by military needs, supported by military funds, and worked in coordination with the Soviet military. Someone with more expertise than I can tell us exactly how military any particular part of the Soviet space effort was (for example, rather a particular bureau was part of a formal military structure or an academic/scientific group that was working with the military, but it is a fact that these partially military program(s) made huge strides, often ahead of the US, that shocked the West and enhanced the USSR's prestige, technical strength, etc. etc. That the demands of the Soviet military often clashed with the more scientific/academic goals of the prominent Soviet scientists does not mean that these strides were not in part attributable to the Soviet military. To name a few:
- First ICBM
- First artificial satellite in orbit
- First human in space
- First human to orbit the earth
- First pictures of the far side of the moon
- First interplanetary probes
Two, direct Soviet military aide was given via advisers and equipment to prop up and defend allies and satellite states.
- Support for the CCP in defeating the US backed Kuomintang in the Chinese Civil War.
- Direct support for the DPRK in the Korean War (including direct support via Mig fighter jets piloted by Russians), a war which forced the US/UN forces into a stalemate
- Deployment of nuclear missiles in Cuba which eventually resulted in US withdrawing it's nuclear missiles from Turkey.
- Advisers and equipment support for the victorious North Vietnamese during the Vietnam war.
The pattern here is that the Soviet military in peacetime (if we can call risking nuclear Armageddon and fighting proxy wars "peace") was employed to spread and defend communist governments and enhance the soviet unions strategic deterrence, rarely through direct military activity (though on rare occasions it was!) but through advisers and equipment. By doing so, the Soviet Union gained friendly markets for exports/imports, friendly votes in the United Nations aligned with it's interest, military allies to strengthen its strategic deterrence, etc. etc. Of course these were more than just military achievements (they were diplomatic as well) and of course in some cases these allied states turned on the Soviets (as in the Chinese) but they were at least short term victories if not always long term.
I am not sure to understand the question, but one possibility could be: - How did the armies of both sides impress the other by their success against pro-Western or pro-Eastern ennemies?
First, the Korean War was a good news for the Soviet: their pilots, on homemade Mig-15, were able to fight the US Air Force and the Chinese, on the ground, effectively fight UN's troops, despite enduring heavy casualties. This situation was a factor of proudness for the Soviet army, plethoric and objectively better in hardware and training than the Chinese army.
Then came all the Israeli-Arab wars, were the Soviet hardware in Arab hands get all the time beaten. But often, Soviets thought that the Arab difficulties in training and command were enough to justify the defeat. The Vietnam war and the initial success of their DCA in the Kippur war lead the Soviets to be very satisfied of their doctrine. Basically, it consisted in: - an heavy assault aviation for first line operations - plethoric and mobile DCA to protect the offensive groups (GMO/OMG)
The Vietnam war were, as Korea, an occasion for Soviets to think: "Our allies, with our materials, beat the US Army so we will beat the US Army". But they missed in the process the numerous ameliorations, on tactical and operative plans, that US forces implemented as a reaction to their defeats.
Note that Soviet navy only got pointless indications during these events, such as the fight between light missile launcher aircraft, while the US Navy reinforces its training and its fighting aviation aboard the carriers. Even the Falkland Wars did not involve proper doctrine or hardware to be very relevant for Soviet navy. Air forces of both sides got numerous the lessons from these proxy wars.
Finally, the Iraq-Iran war showed to the Soviet, providing hardware to Iraqian army, that it was efficient as a robust and easy to use hardware. But they had to note that the US-armed Iran Air Force was able to effectively repel Iraqi hardware, until French Mirages entered the game.
Finally, the very important difficulties of Soviet Army during the War in Afghanistan were a way for the NATO to increase the importance of the special forces and light weapons like mortars, antitank missiles or Manpads. It also showed the disintegration of the Soviet Army, and USSR in general.
To conclude, the proxy wars were factors of gladness and deception for both sides, but notably the NATO sides, especially the US Army, got very important lessons in the proxy wars, even defeat like in Vietnam. The British special forces, aviation and navy got infos in the Falkland Wars, and the French army in its multiple engagements in Africa, against Soviet armed-movements.