Can I setup a PS3 to use a PS4 Dual Shock 4 controller without a PS3 controller?

Yes, you can also use a DualShock 4 wirelessly with a PS3.

From Eurogamer - How to use a DualShock 4 wirelessly with a PS3

Here are the steps:

• Plug in DualShock 4 controller via USB.
• Go to Accessory Settings and Manage Bluetooth Devices.
• Scan for Bluetooth Device.
• Unplug the DualShock 4, press Share and PS Buttons.
• Once device is found, plug the DualShock 4 back in and select Wireless Controller.

Congratulations! You now have a rumble-free DualShock 4 working with your PS3!

Yes, it is possible. According to Business Insider you can plug a PS4 controller into a PS3 with this method:

To use a PlayStation 4 controller with a PlayStation 3 console, the easiest approach is to simply connect the controller using a USB cable. Once the cable is connecting the two pieces of hardware, you'll be able to play PS3 games with the controller right away.

Just keep in mind that there may be a few backwards compatibility issues when connecting the accessory.