Heat Sink compound and older desktops and workstations
Solution 1:
I have experienced this exact same issue with similar Pentium 4 computers from HP. Within about 9 months we reapplied thermal compound to nearly all 80 or so of these PC's. They were all about 4 years old at the time.
After the 5th pc or so, our purchasing department leaned on the hp sales guy and he bought us a case of thermal compound that came in little stickers. The funny part is he ordered 10 times too many so we had these stickers hanging around forever. I ended up doing all of my home PC's, too.
We ended up making a weekend day of servicing every pc of this model. Never had a problem again except for the occasional seized fan.
This was a typical office building. No chemical funny business going on nearby.
Solution 2:
It could be that HP cheapskated when they applied thermal paste to those specific models - who knows after all these years. Like I said in the comment, those processors were notoriously ineffective and hot, which didn't get any better until the Core 2 series.
All your other questions are going to attract too many different opinions for a Q/A site, so I'd strip them out instead of having this question closed, and rather visit us at the Serverfault Chat room to discuss it.
Solution 3:
The thermal interface in the OEM cooling systems has not been "gooey" (even when brand new) for a long time now. But it does degrade over time. Those systems should be at least 7 years old now which is way more than the life span they were designed for - so it's hardly a design flaw/manufacturer's fault.
If you want to keep those running - just replace the thermal interface. But there's nothing to complain about