Short answers, pulled from the story.
The 2nd Panzer Army received its official name on the 5th of October 1941. This designation followed its earlier existence as Panzer Group Guderian which formed on the 5th of June 1940.
General Heinz Guderian commanded the force from its inception on the 5th of June 1940 until his dismissal after the Battle of Moscow. Generaloberst Lothar Rendulic took command by the 30th of November 1943 during the Yugoslav campaign.
Panzer Group 2 implemented the criminal Commissar Order during Operation Barbarossa. In September 1942, the 2nd Panzer Army participated in war crimes while conducting anti-guerrilla operations that killed at least a thousand people and deported over 18,500 individuals.
In August 1943, the 2nd Panzer Army transferred to occupied Yugoslavia under Army Group F. The unit engaged anti-partisan operations against Chetniks led by Draža Mihailović and communist Yugoslav Partisans under Josip Broz Tito without gaining clear victory.
Hitler ordered the detachment of the 2nd Panzer Group to turn southward towards Kiev resulting in an encirclement that trapped 665,000 Soviet soldiers. This operation occurred after forces captured Smolensk and took around 300,000 prisoners.