For instance, Oedipus claws out his eyes after realizing he married his mother. That seems like an odd reaction by modern standards. Was there a culture of self-harm at the time?
Yes and no. Women certainly practiced a lot of self-harm as part of mourning rituals. In particular, they were expected to tear their hair, clothes, and skin while in mourning. Men, not so much. Part of what is so effective about Oedipus stabbing his eyes out is that a man is doing that, and it shows his reversal of status from exalted monarch to criminal outcast (the point of the discovery that he did it also means he has to leave Thebes in exile).
Keep in mind, these examples are from tragedy, which is a literary and performance form specifically designed to be emotional, unusual things on stage. The characters tend to be pushed to places most people wouldn't go.
The classical Greeks (i.e. Sophocles' peers) didn't tend to approve of suicide, but it became more popular in the Roman era, mostly as a way to reclaim or preserve honor.
Oedipus claws out his eyes after realizing he married his mother
He did a lot more than just marry his mother, he had four children by her: Eteocles, Polynices, Antigone, and Ismene. (Though, Euryganeia is also sometimes considered to be the mother of his four children.)
The main characters in Greek stories and plays are 'Heroes'. Their schtick is that they're larger-than-life; they fight extremely well, talk extremely smart, lose extremely big things, and feel extremely strongly.
For example, Heracles impregnated 100 virgins in one night, brutally murdered his entire family, and then embarked on twelve superhuman tasks to repent for it.
Oedipus is a hero, so he reacts in an extreme way to disturbing news.