I assume you are upgrading an Optiplex GX270 MT (since if the only variant which is a tower and supports full size graphic cards)

First, what is your GX270 specs? it usually comes with a wide range of P4s

The Dell Optiplex GX270 motherboard is based on the Intel 865G chipset with 4 DDR slots (up to 4x 1 GB DDR400),2x IDE ports, 2x SATA Ports, 1x AGP x8 slot, 4 PCI slots, supports all FSB 533 and 800 Northwoods (HT as well), some mobo revisions might support prescotts too, the Radeon 9800XT can be a nice upgrade with a good/fast P4, but also you can find other options like Geforce FX 5900/5950, Geforce 6600/6600 GT, Geforce 6800 series, 7300 GT or even 7600 GS/GT

I own several FSP PSUs and none didn't fail on me yet (usually came from OEM PCs, but also have some retail units). Also make sure the PSU really fits on the backplane (despite it uses standard ATX PSUs, it's backplane might not fit with many PSU's AC connector) or you'll have to cut part of the backplane, as for the PSU, it can work with that 9800 XT or almost all the cards i mentioned

I have a few GX270 SFF boards. They definitely work with Prescotts, or at least the boards I've tried it with do. The full desktop version of the board probably supports the same.
I have read of some people running into issues with some models of high clocked Prescotts though. From the troubleshooting threads I've seen, it looks like the GX270 BIOS will detect some older steppings (which have higher power consumption) at high clock speeds (I think at 3.2GHz and above) and will underclock them at POST by reducing the FSB clock.

I would think that 350W PSU should work great, but I don't have experience with newer model FSPs. I think I've run a GX270 board from an old model 300W FSP PSU before, but when I built a real system out of that board I used something bigger. Not sure it was necessary though.
Looks like they mixed up the 5V and 5Vsb ratings on that page, but with 15A on 5V it should be plenty. The 12V is split into 2 rails but I don't see that being a problem either. One consideration is that the ATI 9800 series cards were the last generation to primarily draw their power from 5V instead of 12V, but I expect the 5V@15A rating should still be adequate.

If you care about H.264 HD video playback, some of the late AGP ATI cards from HD2xxx-4xxx are an option. I use an HD2600XT, which will accelerate video nicely in the offline player MPC-BE, but it will not accelerate flash video in a web browser. I think the HD2400 has the same capability for video, not sure how it compares to a 9800XT in gaming though.
I tested a 7600GS in the GX270 and it performed very well, but no H.264 acceleration. Surprisingly, in FlatOut and I think FlatOut 2 the 7600GS was faster than the HD2600XT, but in shader heavy games (which you might never play on a P4) the HD2600XT of course was faster.