Short answers, pulled from the story.
Scholars debate the exact year, though some point to a devastating plague that struck Athens in 430 BC. The play opens with a city choking on disease, suggesting a connection to real historical suffering.
Ancient Greeks called the work simply Oedipus without the later addition of Tyrannus. Aristotle referenced it as Oedipus in his Poetics noting its structural perfection.
Oedipus encountered an old man driving a chariot who struck him with a scepter so he threw him down from the carriage and killed him. This man turned out to be Laius the true father of Oedipus fulfilling the prophecy about patricide.
Jocasta realized the truth when the shepherd admitted that the child he gave away was Laius's own son. She ran into the palace and hanged herself in her bedchamber upon hearing the confession.
The trilogy containing this play took second prize at the City Dionysia competition. Aeschylus's nephew Philocles won first place that year.