Short answers, pulled from the story.
The name Bellerophon means slayer of Belleros, derived from Greek words for slayer and Belleros. One theory suggests it refers to a Corinthian citizen killed by accident during knife-throwing practice.
Hyginus identifies his mother as Eurynome while Apollodorus names her Eurymede. His father is consistently listed as Poseidon though some narratives claim his biological father was Glaucus King of Potniae.
Bellerophon left Corinth after accidentally killing a citizen named Belleros while practicing knife-throwing with friends. Proetus king of Tiryns sent him to King Iobates in Lycia following false accusations made by Proetus wife Anteia or Stheneboea.
Bellerophon attached a large block of lead to his spear and flew head-on toward the beast lodging the metal inside its throat. The creature's fire melted the lead blocking its air passage and suffocating it.
Zeus sent a gadfly to sting Pegasus mid-flight causing Bellerophon to fall back to Earth and die. Other narratives describe him living out his life in misery on the Plain of Aleion in Cilicia where he was blinded by falling into a thorn bush.