Those look more like capacitors to me. Although the probability of them falling off is very small. Have you tried running the mainboard out of the case and heating it up with a heat gun? Sometimes that works.
I've yet to see a laptop fan "fail". 9/10 it's the dust. If not, take the fan off, pull the blades straight off (gently), wipe shaft clean with Toilet tissue(magical stuff) and put a bit of Vaseline in there. WD-40 does not work well. Update: Also, be very careful with the motherboard. These things are more fragile than glass!
I don't see any reason why not, but I wouldn't really spend too much on it. AGP is an old standard, so you won't be able to put it in a more modern PC later. The P3 should support most cards, but try to get one that doesn't need a 6pin power connector. I wouldn't worry too much about the power supply, although yes, everyone will fuss over it :P
Try wiggling the screen; any difference? If it works after a little while of being on, and the cable is not the issue, it's possible there is an issue with the ATi (AMD) chip, I've had it happen to an old laptop before. Although that should be easy to fix. Apparently sometimes the external monitor runs off a separate chip, on some laptops.
I have worked with these computers before, and from experience I think it is the video card. There was a problem with laptops from around that time, if they used Nvidia graphics chips. (NVS 110M for the D620/D630?) If you have an Intel one, then It's most likely something else. If you have the Nvidia thing, Google "GPU Reflow" and you'll find ways to do it. It usually doesn't last very long. If you want it to last long you should reball the GPU. Unfortunately there's no easy way to do it at home...yet.