If you are not allowed to use any microcontroller or microprocessor then things will be difficult if you insist on using the DS1307. I fail to see the reason why you cannot use an Arduino if a PIC microcontroller would be allowed. The Arduino isn't the only microcontroller platform. It should be fairly easy to change the code and make an output drive a relay to switch on/off your lights at certain times.
Yes that is possible, see this project where they use a DS1307, a display and a PIC microcontroller to make a real time clock.