They sell a VGA cable for the 360 (about $40), and I believe they're coming out with an HDMI cable soon, which can easily be converted to DVI. If you have the composite AV cable for the Xbox, you're definitely going to need to buy a new cable as you're not going to get better resolution out of composite than you already have. If you have the component HD cable, there are third party component to DVI adapters available, but it's probably cheaper just to buy a new cable.
DVI, being a digital signal, will look better on a digital monitor than VGA, which is analog. Not all LCD displays are digital, though.
As for the audio, you should be able to plug in the digital coax from the 360 directly into the Fatal1ty and set the card to decode dolby digital.