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I want to read documents that were used to prove German war crimes in the Soviet Union, incl. crimes against civilians and treatment of Soviet prisoners of war.

So far I found the following:

• 
  Nuremberg Trial Proceedings Volume 3 contain some mention of forced labor from the Eastern Europe.


• 20th day, Friday, 14 December 1945

I also looked at Nuremberg trials project. The search for USSR yields mostly documents related to Yugoslavia (which to my knowledge never was part of the USSR).

Search for Russia results in following documents:

1. Transcript for NMT 2: Milch Case


2. 
   Extract from a medical conference report concerning malaria in Southern Russia


3. 
   Affidavit concerning the SS Einsatzgruppen (Task Forces) operation in Russia, including the extermination of Jews


4. Affidavit stating that the order to execute captured commissars was not carried out under Rendulic's command in Russia


5. Affidavit concerning the non-application of the Commissar Order under von Leyser's command in Russia, and his character as an officer


6. Instructions to SS economic and administrative offices for the transfer of economic administrators in eastern Europe to Germany


7. Regulations for SS Economic and Administrative Main Office administrators in Russia, Poland, Norway, and Serbia, including management of economic enterprises and concentration camps

Search for Ukraine gives following results:

1. Cover letter and an outline of a speech concerning the treatment of civilians in the Ukraine


2. Cover letter and an outline of a speech concerning the treatment of civilians in the Ukraine


3. Memoranda concerning the methods used during the deportation of workers in the Ukraine


4. Memoranda concerning the methods used during the deportation of workers in the Ukraine


5. Cover letter and an outline of a speech concerning the treatment of civilians in the Ukraine


6. Cover letter and an outline of a speech concerning the treatment of civilians in the Ukraine


7. Transcript for NMT 3: Justice Case


8. Memoranda concerning the methods used during the deportation of workers in the Ukraine


9. Memoranda concerning the methods used during the deportation of workers in the Ukraine


10. Letter to the chief of the Industrial Armament Department (in Berlin) concerning the execution of Jews in the Ukraine


11. Brief arguing that the imposition of German law in occupied territories was a war crime


12. Transcript for NMT 4: Pohl Case


13. Transcript for NMT 1: Medica Case


14. Transcript for NMT 2: Milch Case


15. Transcript for NMT 7: Hostage Case

Today the majority of historians agrees that the Germans were way more cruel on the Eastern front than in other occupied territories (France, Netherlands etc.).

If that's the case, where and how can I find the documents from the Nuremberg trial that prove that?

Postwar Soviet history books claim that the Germans burned down every village they invaded (especially in Ukraine and Southern Russia). Why, then, don't I see any Soviet documents on German war crimes (only German ones)?

The Nuremberg Trials Project has so far only provided access to 5 out of the 12 Nuremberg Military Tribunals. From the Introduction page on their website:

The Nuremberg Trials Project currently provides access to the document record for five and transcripts for four of the United States Nuremberg Military Tribunals: NMT 1 (Medical Case: U.S.A. v. Karl Brandt et al.), NMT 2 (Milch Case: U.S.A. v. Erhard Milch), NMT 3 (Justice Case: USA v. Josef Altstoetter et al. 1947), NMT 4 (The Pohl Case: U.S.A. v. Pohl et al.) and NMT 7 (The Hostage Case (USA v. Wilhelm List et al. 1947-48).

For the rest, you'd probably have to visit the Harvard Law School Library in person. But remember that the evidence presented was representative, not every war crime and atrocity committed was actually documented at Nuremberg.

As for your supplemental question,

Why, then, don't I see any Soviet documents on German war crimes (only German ones)?

The answer is simple. You do see Soviet evidence.

For example, you asked about Victor Cherevichkin in a previous question. That evidence was gathered by the Soviet Union and presented at Nuremberg.

If you follow the link in my answer, you'll see that evidence about Victor Cherevichkin was presented at the Trial of the major war criminals before the International Military Tribunal, Nuremberg, 14 November 1945-1 October 1946, Volume 7, p455.

If you look at the transcript, you'll see that evidence was presented alongside other Soviet evidence for German war crimes on the Eastern Front. To give just one further example from the evidence presented by Counsellor Smirnov (Assistant Prosecutor for the USSR) on Thursday, 14 February 1946, in the volume cited above:

"Here are a few instances of wholesale bloody murders carried out by the Germans against entire villages. In Yaskino, a village in the region of Smolensk, the Hitlerites shot all the old men and adolescents, and burnt the houses down to the ground. In the village of Pochinok of the same region, the Germans drove all the old men, old women, and children into the collective farm office, locked the doors and burnt them all alive. In the Ukrainian village of Yomelchino in the region of Zhitomir, the Germans locked 68 people into a small hut, sealed the doors and windows and asphyxiated to death everybody inside. In the village of Yershevo, of the Zvenigorod district in the Moscow region now liberated by our troops, the Germans prior to their withdrawal drove about 100 peaceful citizens and wounded Red Army men into a church, locked them in, and blew up the building. In the village of Agrafenovka of the Rostov region, on 16 November, the fascists arrested the entire male population between the ages of 16 and 70 and shot one man of every three."

• Op. cit., p457

The "note" referred to in the transcript was the:

"... note by the People's Commissar for Foreign Affairs of the U.S.S.R., V. M. Molotov, dated 6 January 1942, which was submitted to the Tribunal as Exhibit Number USSR-51."

• Op. cit., p453

This was, in turn, supported by documentary and filmed evidence, collected by Soviet forces and presented at Nuremberg.

Incidentally, all 42 volumes of the official record of the trial of the major civilian and military leaders of Nazi Germany, titled Trial of the major war criminals before the International Military Tribunal, can be found in many libraries, and are also available to read and download in pdf form from a number of sites, including The Library of Congress and archive.org.