My advice is to work through the sound problem before you work on the TV out since you have a very limited machine with a limited time frame.
I would recommend that you dig into the AC97 sound problem on Linux and see if you can find a work around that will allow it to function. I would do this before I bought a new PCI video card. You can't really do anything without sound. You can start by doing a search on the board for AC97 and there is a lot of info.

AGP video is a lot better and you have a lot more choices. If you can get the sound working then you can work on the PCI video issue....

Good Luck