Short answers, pulled from the story.
The Schutzstaffel received its final name on the 9th of November 1925 following instructions from Adolf Hitler to Heinrich Himmler. The organization evolved from a small volunteer guard unit known as the Saal-Schutz formed in 1923 to provide security for Nazi Party meetings.
SS officer candidates were required to prove Aryan ancestry dating back to 1750 during the early years of the organization. Other ranks needed only to demonstrate Aryan grandparents by 1800 though these physical criteria were loosely enforced for over half the men.
Hermann Göring transferred control of the Geheime Staatspolizei or Gestapo to Heinrich Himmler on the 20th of April 1934. By the 17th of June 1936 all police forces throughout Germany were united under Himmler's administrative control alongside the SD Kriminalpolizei and Ordnungspolizei.
By early 1942 Auschwitz expanded to include gas chambers where victims were killed using pesticide Zyklon B on Himmler orders. Stationary gassing facilities replaced Einsatzgruppen methods using carbon monoxide gas from automobile engines during Operation Reinhard starting in late 1941.
Historian Raul Hilberg estimates Einsatzgruppen and related agencies murdered more than two million people including 1.3 million Jews between 1941 and 1945. The greatest extent of Einsatzgruppen action occurred in 1941 and 1942 within Ukraine and Russia following standard procedures coordinating access to execution grounds.