Bad logicboard/processor, LCD, or video cable?

It's not the MLB if you're getting undistorted external video. That's almost always a sign of a bad LCD panel, though there is at least a small chance it's the cable, and I suppose an even smaller chance the connector on the logic board is bad. I'd put my money on the display panel.

I would also strongly suggest listening to Bob's advice. At 400MHz, that thing has got to be over 10 years old at this point. It's had a good run, and you definitely got your money's worth out of it, but it's time to let it die. If you're really dead set on continuing to use it, then just have it permanently connected to an external monitor. The reason you don't see desktop icons or anything else, is because the external display is being set up as a secondary display, in an extended desktop setup, not as a mirrored display.

At 400MHz, I can't imagine you're using much beyond 10.2, and I have little experience with that so I don't know if right clicking (or Option + Click) on the desktop will give you an option to adjust the resolution so you can set up mirroring instead of extended desktop. Some older models also had a keyboard combo for switching between the modes, so be sure to check for that. You really probably are better off listing the thing on ebay for parts, and hope there's someone out there stupid enough to buy it. Or, I don't believe there are any age restrictions on Apple's trade-in program. You can take in pretty much any old Mac (may not even have to be an Apple product) and then get a gift card for use in the store. Apple will then dispose of the old computer properly. Best Buy has something similar, though I believe it's sans gift card. They'll just take your old e-waste off your hands for you. Whether they just bulk ship it to China where they burn it in big open pits to try and get at the valuable metals in a computer (which is really bad for the environment and the health of the workers and surrounding towns) or do it more responsibly like Apple, I can't say.