Finally, I accidently stumbled on a sort of solution, if you can call it that. What I discovered was that the software that comes with the TV2000XP Deluxe doesn't like when the resolution changes while it's playing the TV. When it does, the TV window goes blank and its sound cut off.
So, to go arround this problem, I simply set my games resolution to be the same as my primary desktop. Therefore, when I enter the game, the resolution doesn't change and the TV keeps on playing on my second monitor. I've had this work with the Savage Demo, Warcraft 3 and UT 2003. I didn't try with Halo yet but since I'm playing it at 800x600 because it's slow otherwise on my Radeon 9700 Pro, I don't think I'll try running it and the TV at the same time.
Anyways, this is just in case someone else has the same problem. If you do, I hope it works in your case as well...
(Btw, I was using an Athlon XP 2200+, 1Gb ram PC2700, Radeon 9700 Pro, DX9, WinFast TV2000XP Deluxe, Windows XP. My desktop resolutions are Primary: 1152x864, Secondary: 800x600 and I wasn't getting much slowdown in Savage running at 1152x864. Keep in mind that a slower computer could have slowdown resulting in playing a game and watching TV at the same time.)