No, your Duo-R PSU will not work with the SGX unfortunately. You'd need to buy a generic one with suitable specs:
9V DC 800mA
(+)----(o----(-)
Outer Diameter: 5.5mm | Inner Diameter: 3.1mm
Didn't even notice the AV cable on that Hit-Japan listing, it's definitely not official. My bet is that it's probably from a Mega Drive or something. It would still work but without stereo sound obviously. That said, you could always use your AV cable from your Duo-R.