We all know now that Stalin's real name was Jughashvili (Ioseb Besarionis dzе Jughashvili). Lenin was actually Ulyanov, Trotsky was Lev Davidovich Bronstein, Kamenev was Leo Rosenfeld, etc.
Searching through literature and internet (both international and Russian) you could find many explanations as to why the Bolsheviks had pseudonyms, how Stalin chose to be Stalin and Koba before that, etc.
However, I could not find information on whether the average Soviet citizen (let's say a factory worker or kolkhoz farmer) knew or could have known the birth name of Stalin or other Soviet leaders, even those who were disgraced like Trotsky. Note that Soviet press and other media constantly praised Stalin and maligned Trotsky after his exile in 1929. But I could not find any information that they ever used the birth names of either of them. And if not the press, did at least Soviet encyclopedias (available to a much smaller circle of academics) mention these details? Were there even word-of-mouth rumors about the origins and birth names of the leaders?
EDIT : To clarify matters further, I will give one simple example. After the capture of Yakov Dzhugashvili, Germans issued propaganda leaflets calling him elder Stalin's son. Could average Soviet soldier (poor peasant or factory worker) make a connection between Dzhugashvili and Stalin, which is of course necessary for propaganda to work ?
Of course it was widely known. In Lenin's case, in 1924 they even renamed his native city Ulyanovsk. For instance, the History of the All-Union Communist Party (Bolsheviks): Short Course, which was published since 1938 and until Stalin's death in millions of copies, for instance, contains a passage referring to Lenin as "Ulyanov."
As for Stalin, his "Collected works" (published in 1946, again in millions of copies) contains a brief bio of the author, which starts with:
9 (21) декабря. В г. Гори (Грузия) родился Иосиф Виссарионович Джугашвили (Сталин).
December 9 (21). Iosif Vissarionovich Dzhugashvili (Stalin) was born in Gori (Georgia).
Edit. Regarding the question in the edit (about Stalin's son): I think it is impossible to tell. (Who would or could collect this type of data? Only NKVD, but they had their hands full with other, more pressing, issues. Even anecdotal references are unlikely since peasants do not write memoirs.) My best guess is that this really did not matter: An average Soviet peasant (and USSR was still primarily a peasant country at the time) could not care less about Stalin's son. (If they hated Stalin, and many did, they would have hated his son as well.) The main initial defining factors (in the degree of support for Nazi occupiers and willingness for soldiers to surrender) would have been how much he/she hated the collectivization plus the tactical situation on the ground. For that, there are some serious historic studies and I could find references (do not remember off the top of my head), maybe it was already discussed at HSE. In the later stages, there were other defining factors, e.g. Nazi atrocities on the occupied territories.