In Downfall, Hitler laments about how he should have purged the military officers like Stalin did, so my question is: would people have actually known about Stalin's purges during that time ?
Of course. It wasn't much of a secret. Some 85% of the Soviet military's top positions were removed; the scale alone makes it rather impossible to hide. In fact, knowledge that Stalin had decapitated his own army's backbone was one of the factors motivating Nazi Germany to invade.
But not only did [the Great Purges] do incalculable damage to the future construction of Soviet military strength; they also instilled in Hitler and his advisers an indelible notion of the weakness of the Red Army. To Hitler, this very weakness was an invitation to strike before a powerful military machine could be constructed. In Hitler's eyes, then, Stalin's purges gave him the chance. He thought Stalin must be mad.
- Kershaw, Ian. Fateful Choices: Ten Decisions that Changed the World, 1940-1941. Penguin, 2007.
Yes there were many trials. The extent was not nesscarily known but the removal of many senior officers was well known.
This is a very good question. The answer is: yes. Everyone who wanted to know, knew. (This includes Hitler). There were plenty of people who managed to escape to the West, they published their accounts. There were plenty of other sources: Soviet union was never closed completely.
On the other hand, enormous pro-Soviet propaganda, supported by many left-wing Western intellectuals (some of them simply bribed by the Soviet regime) strongly influenced the public opinion. As a result the "general public" did not know much. The Soviets used assassination and intimidation to silence those who wrote on the subject in the West.
By the way the same happens now: Russia continues to manipulate the media of the West to a large extent. I mean the Russian aggression in Ukraine, of course, but not only that. Everyone who cares knows about assassination of Litvinenko, who exposed Putin's regime, for example.
(Downvoting of this answer is another proof of what I wrote, by the way).
Yes, he knew, and almost anyone who cared knew - see all the answers before this one
From many sources we know that Germany actually actively used Stalin's paranoia to "provide" some damaging information alleging that some generals in USSR's High Command (Tukhachevsky et al.) conspired against Stalin - therefore "helping" Stalin to weaken the Soviet army
Hitler's lament is mostly infantile and self-serving. First of all, one of the reasons that Hitler even dared to invade USSR was that he himself and his military command judged the Soviet Armed forces significantly reduced in their preparedness and overall might by these purges. Second, level of resistance to Hitler in his army was still extremely low even after 1944 overthrow attempt and deterioration of Germany's standing in the WWII at the time. It is well known that quite a few of his generals complained about his many incompetent decisions in the military matters -- but still went along with them as it was nearly impossible for them to defy his orders. So his "lament" basically only served his need to move the fault to someone else.
However, there is some evidence to the following
a) some German generals were not great military commanders who failed because they followed Hitler's "stupid orders"
b) some in the Germany High Command pursued their own agenda, which sometimes undermined Hitler's plans - but general consensus says that does not seem to have played a major role in Germany's defeat
Williamson Murray and Allan R. Millet, A War to Be Won: Fighting the Second World War, The Belknap Press of Harvard University, Cambridge, Massachusetts, 2000
Geoffrey P. Megargee, Inside Hitler's High Command. University Press of Kansas, 2002
Leopold Trepper, The great game: memoirs of the spy Hitler couldn't silence. Нью-Йорк: McGraw-Hill, 1977