DVI doesn't include audio signal, so if you're connecting DVI out on your computer through an adapter to HDMI in on your TV, it won't bring sound along.
Sort of. You can't typically use HDMI with an external sound source on most TV's. Since HDMI typically carriers sound and video the TV doesn't look for another source of sound. So, unless you have a way to get the TV to use component sound in and HDMI video you're not going to get sound.
You might try finding a DVI to component (RGB) cable and then hooking up your sound card/on board sound to the same set of component connections. Otherwise you'll need a video card with HDMI out to get sound.