Does installing iOS 7 disable Find My iPhone?
No, Find My iPhone will remain activated. Using Update to update the device will keep your settings, including the device's Find my iPhone status.

If you put your device in recovery mode or DFU mode then once the restore has taken place the activation lock will be enabled and you will be prompted to bypass the Find My iPhone with your Apple ID.

However, if you attempt to restore the device in iTunes, you will be prompted to deactivate Find My iPhone before the restore can take place.