xboxdrv can do that, but only when the PS3 controller is connected via USB:
As for the ids, I listed them below, but they won't really help you much. As you can see, Xbox360 USB and Xbox360 Wireless use different ones and most games don't care either way, they just use the button one after another and don't care which button is mapped to which physical button, you have to remap them manually in each game.
As for Wine, Wine only supports DirectInput and lacks Xinput support and many modern games require Xinput. To get around that you have to use a Xinput emulation .dll (same trick you have to use on Windows with older gamepads), haven't tried it myself, but it's described here:
USB Xbox360 Controller
Wireless Xbox360 Controller