1. all call upon the healer apollo. what does apollo have to do with
healing? why should oedipus consult this god's oracle? why does apollo
have an oracle?

Apollo is indeed the God of Healing. Since he is the God of Light and Life, healing is associated with him. He is also using sun as a symbol together with bow.

Oedipus used Apollo's oracle which is located in his temple in Delphi to ask for Apollo's advice regarding unexplained plague which was occuring in Thebes that time. Phytia answered him. He asked his brother in law, Creon to see the plague. When he returns, he told Oedipus that it is a punishment from Gods. The Oracle also revealed a prediction that Oedipus will kill his father and will be married to his mother.

Apollo has an oracle because he uses it to communicate to people. He is using an Phytia, one of the priestess in the temple to send his message to everyone. His message can be composed of advices, future (predictions), questions and guidance to people, the Delphi city and other things.