Short answers, pulled from the story.
Mikhail Mishustin is the current Prime Minister of Russia. He was first appointed on the 16th of January 2020 after Dmitry Medvedev and the rest of the government resigned the previous day, and was confirmed for a second term on the 10th of May 2024.
The modern post of Prime Minister of Russia was established by a decree of Emperor Nicholas II on the 19th of October 1905, which created the Council of Ministers and made its chairperson a fully-fledged head of government. The current constitutional form of the office dates from the 12th of December 1993.
The Prime Minister of Russia is nominated by the president and must receive the consent of the State Duma. The Duma has two weeks to vote on the nominee. If the Duma rejects the president's candidate three times consecutively, the president may dissolve the Duma and appoint a prime minister without its consent.
If the president of Russia dies, resigns, or is impeached, the prime minister becomes acting president until new presidential elections are held within three months. As acting president, the prime minister may not dissolve the State Duma, call a referendum, or propose constitutional amendments.
Boris Yeltsin served as the first head of government of the Russian Federation after the Soviet Union's dissolution, though he held the role as president rather than prime minister. Viktor Chernomyrdin was formally appointed the first prime minister in the modern sense on the 14th of December 1992, after acting prime minister Yegor Gaidar was refused approval by the Supreme Soviet.
The president of Russia appoints and dismisses the prime minister, may personally chair cabinet meetings, and can issue binding orders to the prime minister and other government members. The president may also revoke any act of the government. The prime minister and the entire government are constitutionally required to resign after each presidential inauguration.