Short answers, pulled from the story.
The photograph was taken by Soviet photographer Yevgeny Khaldei on the 2nd of May 1945. He shot it with a Leica III rangefinder camera with a 35mm f3.5 lens after climbing the pacified Reichstag building in Berlin.
According to Khaldei's own account, the soldiers were 18-year-old Private Aleksei Kovalev from Burlin in the Kazakh Soviet Socialist Republic, Abdulkhakim Ismailov from the Dagestan Autonomous Soviet Socialist Republic, and Leonid Gorychev from Minsk. The Soviet official story named Meliton Kantaria and Mikhail Yegorov, but their identities were disputed for decades.
Yes. Khaldei altered the image before publication by removing a second watch from Senior Sergeant Ismailov's wrist using a needle, to avoid any implication of looting. He also increased contrast and added smoke from another negative on the same roll of film, making the image more square in format.
The photograph was published on the 13th of May 1945 in Ogonyok magazine. Many other photographers had taken images of flags on the Reichstag roof, but Khaldei's version became the defining image.
The copyright belongs to TASS, the Soviet news agency Khaldei worked for when he took the photograph. Russia's copyright term for works by legal entities is 70 years after publication, so the copyright expired on the 1st of January 2016. A 2015 court case brought by Khaldei's daughter Anna confirmed TASS held the rights, not the photographer's family.
No. The Reichstag had been closed for 12 years by the time Soviet forces captured it on the 2nd of May 1945. Following the Reichstag fire of 1933, all legislative sessions moved to the nearby Kroll Opera House. The Nazis themselves viewed the building as a symbol of democratic weakness rather than their own power.