Personally I book appointments for as late in the day as I can and just close early. I would never take a child with me other than my own. If the appointment is out of my city then I just book the full day off. I would never open late because I never know how long I will be so hard to tell parents to come at a certain time when you never know when you will be back. At least if you close early it doesn't matter how late you will be at the appointment.
My contract states that if I close at noon I only charge half day's rate. If I close 3 or later I charge full rate. I would never close during nap time.