To pull out the timeline of these examples, the Great Purge was 1937-1938, Lenin was the leader of Russia from 1917 to 1924 and of the USSR from 1922 to 1924, his death. You are perhaps referring to the November 1917 Constituent Assembly election, where the Socialist-Revolutionary party led with around a third of the vote and the Bolsheviks secured around a quarter of the vote (several other minor and regional parties compose the rest of the bloc; in Yakutia, for example, the Bolsheviks weren't even on the ballot)--and the following year the Bolsheviks seized power. After the Kronstadt Rebellion political factions themselves were banned, and the fledgling USSR functionally became a one-party state. The DDR did not exist until 1949, and the Berlin Wall was not built until 1961. Although some scholars draw out the revolutionary character of the Soviet Union, others point to concrete endings of the 'Revolution' in the early 1920s, at the rise of Stalin, or the onset of the second world war; yet others push back the origin of the Revolution to 1911, 1905, or even earlier as a late nineteenth century low-intensity civil conflict. A narrow chronology of the Russian Revolution proper would be 1917-1922.

Many dozens of scholars have composed the Western view of the Soviet Union, and similarly dozens have sought to dissect or challenge these views. A historiographical exploration of Western perceptions would make a great book or three, so I'm afraid it's too capacious for this forum, but I will include a brief reading list as a good starting point.

To take your first example, did Stalin execute his enemies? Yes. To extend this more generally, political executions and exile was common under Stalin's reign, both at high-level political infighting and in the hundreds of thousands executed in the Great Purge. Stalin personally approved hundreds of these executions, and his immediate aides approved tens of thousands; the delegation/direction question is a point of contention, and like many historical questions is quite complex.

The last Moscow Show Trial, the 1938 Trial of Twenty One, is the most prominent example of the execution of Stalin's enemies: Nikolai Bukharin, a prominent early theoretician of the Bolsheviks and general secretary of Comintern in the late 1920s, was executed for (real and imagined) disloyalty, 'Trotskyism', and wrecking. Genrikh Yagoda, the former NKVD director who ironically had supervised the executions of Lev Kamenev and Grigorii Zinoviev, was himself found guilty and executed. Alongside Bukharin, former Chairman Alexei Rykov, and Secretary under Stalin Isaak Zelenskii, and other politicians prominent in the 1920s were charged and executed.

The propaganda question is interesting. Immediate reactions to the show trials, for example, were generally poor. The American Dewey Commission of 1937 conducted hearings in Mexico, where Trotsky was living in exile, determined that the show trial defendants were innocent. The political inclinations of the members varied greatly, from anti-communist to socialist to Trotskyist. Some Western sympathizers continued their support for the USSR, and of course a number of Western anti-communists did perceive the show trials as one affliction out of many. But the virulent anticommunism and antisovietism that flavors Cold War era perceptions are rather removed from those of the 1930s, and I am not sure how much continuity I can see given Stalin's death in 1953 and the Soviet break with Stalinism in 1956.

There was and is shoddy scholarship on the Soviet Union based on political leanings, as well as excellent scholarship from across the political spectrum. Few students of the Soviet era walk away avowed communists, but this doesn't mean that the 'full picture' is necessarily missing: just as we needn't give equal footing to all political claims today, we can dismiss some claims out of hand.

We have, for example, Stalin's initials on so-called 'kill lists': there is not much ambiguity to the minimal role Stalin might have played in the Great Purge. Instead, the debate is on how much of an active role Stalin occupied. Similarly, the existence of the Holodomor and central Asian famines is not contestable: we have first-hand writings and descriptions, census data (that was itself censored), photographs, archaeological evidence, and oral records. These famines inarguably occurred. Instead, the questions that are debated circle around how active of a role the Soviet government took in perpetrating the famines, what caused the failure to ameliorate them, and so on.

'America’s understanding' is a difficult idea, because American academics compose a pretty hefty share of these debates, while the government's position has changed considerably over time (and, of course, varies from agencies and actors). A great amount of Soviet studies were government supported for security interests, which does impact how we should read Cold War era scholarship, but the nuances there are far finer than "American propaganda about how Stalin’s political enemies disappeared".

Robert Conquest, J. Arch Getty, Robert Thurston, Anne Applebaum, and Norman Naimark are good scholars to draw from, with some competing views on the intentionality behind Stalin's actions as well as how direct of an influence Stalin had throughout the 1930s and 1940s; nonetheless, the claim that Stalin oversaw the execution of hundreds of thousands of Soviet citizens is not challenged by most historians.

So I'll mostly answer this by linking to a few things.

American understanding of the Russian Revolution and the Soviet Union in general was certainly shaped in a lot of ways by the Cold War, but this was as much by a lack of access to information as to outright propaganda.

More as to how academic understandings of the USSR developed in the West, and particularly in the US, can be found in an answer I wrote here.

I also did an answer as to how Soviet elections worked here.

I also did an answer on Soviet living standards (and how East Berlin does and does not fit into arguments on that topic) here.

I guess I would say overall though is to be very wary of people making these arguments. They tend to be based off of cranks and denialists like Grover Furr, and tend to take extreme contrarian positions - if the US popular viewpoint is influenced by the Cold War and says that the USSR was a horrible place*, then the complete opposite must actually be true, and all the horrible things are actually lies and US propaganda.

* Soviet horribleness is actually part of the historic debate in that first answer I linked to, and I think it's kind of a nuanced question. Because clearly some very horrible things happened under Soviet rule, but often Western interest in the USSR seems to begin and end with gulags, famines, nukes and spies, and that very much is an interest shaped by the Cold War. But the USSR (both state and society) in the 1960s - 1980s was a very different place from the USSR of the 1930s-1950s, even if it was still very isolated and authoritarian.