URC7965: Do I have to plug the PS3 Adapter into the PS3 via USB?

No, this is not necessary as the Adapter will run fine using batteries alone.
However if you do plug it into your PS3 (using a standard USB to Mini USB cable like the one used to charge the PS3 Controller), the Adapter will draw power from the PS3 when it is powered on, extending the life of the batteries.