Did the Sphinx kill himself when Oedipus got a riddle?

No, the Sphinx did not kill himself. In the myth of Oedipus, the Sphinx was a creature who posed a riddle to anyone who passed by Thebes, killing those who failed to answer it. According to the myth, Oedipus was the only one who was able to answer the riddle and thus broke the Sphinx's curse. There is no mention in the myth of the Sphinx committing suicide.