Is there an angel charged with earthquakes, who are the two angels that turned Lot's village upside down, and what is the mission of Gabriel, peace be upon him, after the death of Prophet Muhammad, peace and blessings be upon him?

There is no text stating that there is an angel tasked with earthquakes. The angel through whom Allah destroyed the people of Lut was Jibreel (peace be upon him), for he overturned their cities with his wing. We only know of Jibreel's tasks what has been mentioned in the infallible texts. Among these is that he descends with the angels on the Night of Decree and greets the Muslims. Jibreel may also be present at the death of a pure believer.