Is it possible? Does someone just need to write the bluetooth drivers for the Droid or what?
I don't think it can yet, I'm no expert but as far as I know from what I've read, the PS3 controller uses bluetooth HID which isn't supported on the android OS -yet. Only bluetooth SPP works for controllers at the moment.
Unfortunately, no there is not - the USB cable is necessary to both charge the controller and pair it with the console. Most major retailers and specialty websites should carry a number of replacement cables, and (like other posters have mentioned) you can sometimes even find brand-new controllers available that include additional charging cables.