Short answers, pulled from the story.
August Rei declared the official Estonian government-in-exile on the 12th of January 1953, in Oslo, Norway. Oslo was chosen over Stockholm because Norway did not ban such political activity, while Sweden did.
Sweden had laws banning political activity of the kind the Estonian exiles intended, while Norway imposed no such restrictions. That legal difference made Oslo the practical choice for the formal declaration on the 12th of January 1953.
Enno Penno was the final acting prime minister of the Estonian government-in-exile, serving from the 1st of March 1990 until the 15th of September 1992. He formally ended the government's work when Heinrich Mark handed credentials to incoming president Lennart Meri on the 8th of October 1992.
No. Under the American Stimson Doctrine, the United States never recognized the legitimacy of the Soviet occupation of the Baltic states, including Estonia. This policy gave the Estonian government-in-exile a degree of international standing throughout its existence.
Tief's government briefly seized control of Tallinn on the 21st of September 1944, raising the Estonian tricolour in the Pikk Hermann flag tower, but Soviet forces retook the city on the 22nd. Most government members were captured, jailed, deported, or executed. Tief survived a decade in Siberia and died in Estonia in 1976.
Five governments-in-exile were formed in succession between 1953 and 1992, each led by an acting prime minister who assumed the role by constitutional succession. Heinrich Mark held the position longest, from the 8th of May 1971 to the 1st of March 1990.