Short answers, pulled from the story.
Japan and the Soviet Union signed the Joint Declaration on the 19th of October 1956 in Moscow. This document officially ended the state of war that had persisted since World War II.
Article 9 of the Joint Declaration stipulated that the Union of Soviet Socialist Republics agreed to transfer the Habomai Islands and the island of Shikotan to Japan. The actual transfer was scheduled to take place after the conclusion of a formal Peace Treaty.
No peace treaty has been signed because the United States threatened to keep the Ryukyu Islands if Japan gave away other territories. American pressure prevented the negotiation of the promised comprehensive peace settlement, leaving the core dispute unresolved.
Yes, the Soviet Union waived all claims for World War II reparations from Tokyo within the text of the declaration. These concessions removed a major financial burden from post-war Japan and facilitated the normalization of diplomatic relations.
On the 14th of November 2004, Russian foreign minister Sergei Lavrov stated in an NTV interview that the Russian Federation recognized the Declaration of 1956. He declared that Russia was ready to have territorial talks with Japan on that basis.