i have dvi connectors on my video card (470gtx) and am using a dvi to hdmi cable. i'm getting good sound without any issues. i had sound on my tv speakers and now that i hooked the cable up to my receiver i have sound through my theatre system. if you search google it appears that others are saying what you are about dvi carrying no sound but when such a cable is used people are getting sound. strange indeed but worth mentioning.
as suggested, i would use a video card which supports two monitors. if you have native hdmi out then this would be best. if not then you can get a dvi to hdmi conversion cable. if for some reason you don't have two ports or a video card which supports two monitors there are video splitter cables or switch boxes available. keep in mind that your monitor's native resolution and the tv's native resolution should match if you don't want any issues (when you direct copy desktops).
if you just want to use your monitor as an "extended desktop" you can use the setting to extend your windows desktop onto this monitor and just drag what you want to see onto the screen. when i hooked up different resolution monitors this is what i did. i experienced less issues this way.