Short answers, pulled from the story.
Reinhard Heydrich formed an ad hoc unit in March 1938 to secure government buildings and documents following the German annexation of Austria. This group operated under the Security Police before evolving into the Einsatzgruppen that would later murder millions.
Adolf Hitler ordered the invasion of Poland on the 1st of September 1939, prompting Heydrich to re-form the Einsatzgruppen to follow the German armies. Werner Best led the re-formed units for the invasion of Poland while Bruno Streckenbach, Emanuel Schäfer, Herbert Fischer, Erich Naumann, Udo von Woyrsch, and Otto Rasch commanded specific battalions attached to various army groups.
The largest mass shooting occurred at Babi Yar on the 29th and the 30th of September 1941 northwest of Kiev city center. The murders continued for two days claiming a total of 33,771 victims who were shot in the back of the head or neck after being herded down into the ravine.
Himmler found shooting methods inefficient and demoralizing for troops after witnessing an execution in Minsk during August 1941. Many perpetrators suffered physical and mental health problems and turned to drink as a coping mechanism so he decided to transition to gassing victims especially women and children.
After World War II ended, 24 senior leaders of the Einsatzgruppen were prosecuted in the Einsatzgruppen trial held from 1947 to 1948 under United States military authority. Fourteen death sentences and two life sentences were among judgments rendered during these proceedings while only four executions were carried out on the 7th of June 1951.