To get audio from the DVI port on your video card (assuming your card supports that) you’ll need the adapter made specifically for that card, a standard DVI to HDMI adapter won’t work since audio isn’t part of the DVI standard, so different manufacturers have different schemes to make that work.
Beyond that, I’d think it should be possible to have both outputs active (as if you have two monitors), but it would probably take some tweaking of settings.
