AFAIK the digital video signal used in DVI and HDMI is identical. such that you could simply get a straight through converter to change between one and the other.
The difference is DVI also has the ability to carry analog video as well as digital video. And as you said HDMI carries digital audio as well as digital video.
Personally I like using DVI and an optical audio connection over HDMI. My Surround system is completely separate from my display so it doesn't make much sense sending my audio and video in the same cable unless it was going into a receiver that would break out the signals for me.
With that said, looking at the REAL specs, it doesn't seem like that display supports any digital inputs... it's HD support is limited to VGA/RGB and component.
The Xbox360 wont support digital at launch but you'd be best to future-proof your display by making sure it does.