Short answers, pulled from the story.
Medieval Arabic texts called them Bjnak, Bjanak, or Bajanak while Classical Tibetan documents recorded their name as Be-ča-nag. Byzantine authors like Anna Komnene referred to them as Patzinakoi and East Slavic peoples identified them as Pečenegi or Pečenezi.
An Uyghur envoy reported armed conflict between Be-ča-nag and Hor peoples near the Syr Darya region before 850 which triggered a westward migration across the Khazar Khaganate. They invaded Hungarian dwelling places along the Kuban River and upper Donets course around 830 or during the 850s after being forced out by Karluks and Oghuz Turks.
Combined Byzantine and Cuman armies annihilated them as an independent force in 1091 at Battle of Levounion. Emperor Alexios I Komnenos recruited survivors who settled in Moglena district today within Macedonia following this defeat.
Byzantine emperor Constantine VII listed eight tribal groupings four on each side of Dnieper river that divided further into 40 sub-tribes likely representing clans. Each name consists of two parts combining equine coat color plus ruler's title such as Yavdı-Erdim meaning tribe with brilliant horses.
Some historians believe Pechenegs were ancestors of modern Karakalpaks through migration to the Aral region after 11th, 12th century defeats. Consolidation of Pecheneg tribes in southeastern Aral region during 9th, 11th centuries played role in Karakalpak formation via shared traditions and nomadic lifestyles.