If I'm not mistaken, I believe that "time of day" was probably figured based off of "noon". Noon is, literally, mid-day... the point at which the sun is at it's highest point in the sky. The "clock" begins again at the next hour... hence, one O'clock in the afternoon.
If noon is mid-day, it would make sense that mid-night would halfway between the prior noon and the next.
As for why it is divided into 24 - 60 minute hours... I have no idea.
How did it come to be that sunrise and sunset was decided to be 6ish?
If sunrise/sunset are around 6, isn't midnight the middle of the night? It diverges somewhat because of sunrise/sunset varying with season and with Daylight Saving Time, but your text acknowledges that midnight is the middle of the night.