I thought there was a gold dot under each prong, save for those missing...
If it's like this it's oK. the prongs are supposed to be in contact with such gold dots so the current can flow. If there is no dot, there is no need for a prong in that place. All socket-775 CPUs have the same dot scheme underside, so if the celeron works, all other will (as long as the mobo supports them anyway)
A quickie about the compound
Stop a second. Look at
this video to learn how to do it right.
Also,
always buy a thermal compound that is non-conductive and non-capacitive (it should be clearly stated). This way if some of it oozes around it won't short out contacts.
I went to Frys, but became discouraged b/c all their socket 775 models seemed so huge, with what looked like brass tubing
The "brass tubing" you see is actually a far more high-tech
heat pipe. It carries around heat (much) more effectively than a solid lump of metal like the current one you have and allows them to design heat rejection systems that are far more effective without going to full watercooling.
What matters is the bottom part and the connection brackets. If that is the right size and locks in place, even if it looks funky it will do its job.
Just make sure it fits in the case, as some are designed for far bigger gaming cases.
Finally, with a chip change, I was reading offhandedly about how the BIOS settings have to be "reset?"
It's more a precaution though, in case someone was overclocking things or enabled whatever feature the other one doesn't have (although the BIOS should usually detect and auto-adjust).
Anyway, the board has a jumper to do it fast and easy, look at the page I linked above, the last entry called "clearing BIOS settings", and then "clearing CMOS"