I think it's possible with Bluetooth and dualshock yeah, but in my experience getting those things to work is a pain in the arse. For different reasons it might be difficult or easier on Linux. It usually comes down to buying an exact usb dongle listed on a wiki somewhere and then trying a driver someone wrote or bundled with the distro.
Small update - I trotted running to power for the Pi from the USB port on the hdmi board rather that from its own power supply... it worked Means I only need on power supply for the whole thing rather than 2 or 3 (speakers should rib of the board to). Only thing is, for the Pi to pick up the screen, it needs to turn on a few seconds after the screen gets power, so in going to get a usb power switch so it'll be turn on the master power > everything is on standby > flick usb power and then the Pi picks up the screen and turns that on too.
I've read some more challenging games can under perform if you don't have the correct amount of power running to the Pi... Is it worth checking that it will get enough from the usb instead of from mains? Ie it might work to load up, but not when it's being really tested.
Can it do all that via USB though? There are some "hi power / super charge" usb ports on recent motherboards but any more than 500mAh is beyond USB 2 spec. They would have to make a point of including the higher current. You need 1mAh (1 amp) to power the Pi fully. It might switch on but weird stuff happens if it needs to draw more power, which or course it doesn't if it doesn't need to, just like a mobile phone.
Adafruit Switched usb lead specifically for the Pi which provides the correct power.
Adafruit Stereo Enclosed Speaker Set - 3W 4 Ohm. I don't actually know if this will work at all, I just know the vga board I have has a 4 pin audio amp on it with that connector so I'm taking a punt and it's only £8. If it works the if saves a load of wiring hassle as these are plug and play (I hope).
With passive speakers into an amp it's quite difficult to break them. The main thing that matters is the impedance output of the amp, that needs to match the speaker so try to find a datasheet for the board. The impedance of the speaker can't be higher than the amp can deliver or they will draw too many current and fry amp.
Most likely worse case scenario is you need to crosswire some of the connections with bits of bell wire until the speakers work. Actually, that may not even matter much as all that happens if the + and - are switched is the speakers are out of phase. Obviously - and - to a speaker and vice versa won't work. For example pins 1 through 4 go - + - + and the speaker connection is wired something different.
4ohm is quite typical for smaller things. And sometimes speakers can operate at 4ohm and 8ohm to use less power, sometimes amps can be switched to use both. But forcing a speaker to draw too little current than it needs can stress the speaker and eventually kill it.