I'm not asking you to unplug the CPU's fan while the system is running!
(That would be a bad idea & the system would over-heat for sure).
The fact that you are not able to enter the bios is a major factor & you should have posted this.
So far this is what I do know. Memory was added to the system.
You can not enter the motherboards bios menu. Your old CPU fan was running at 1713Rpm, temp 46C which is, 114.8 Fahrenheit & you cleaned the CPU fan & assembly.
Than remove the heat-sink & fan assembly & applied new thermal grease.
(Guess the video's helped)?
Now tell me. Did the problem - (System restarts), did this start happening shortly after you added new memory onto the motherboard?
This is what I want you to do. Gather up a couple cotton swabs & a bottle of house hold alcohol. Unplug the power cord from the PC. Open your case. There are two memory slots on the board. You said you added memory 2 years ago. Carefully remove the memory from their slots. Look to pages 102-104 in your user manual for instruction on how to remove & install your memory. If you no longer have your user manual you can download it & print it out from this link: http://h10032.www1.hp.com/ctg/Manual/bpb13287.pdf
You will also need Adobe PDF reader to view the manual, which can be download for free at Adobe's web site: http://www.adobe.com/products/reader/
Once the memory has been removed take a cotton swab & place a couple drops of alcohol on it.
(Not dripping wet). Wipe the memory’s contacts/ each side both sticks. (Do not touch the contacts with your fingers). Use canned air & blow the memory slots out. If you do not have canned air you can use a powered sweeper with the brush attachment to clean the memory slots out. Than re-insert both DIMMs back into their slots. (Press firmly down while "slightly" rocking the memory left to right as you push down until both locking ends snap onto the memory). Before you close the case, I want you to check every connection in the case. Any one data cable or power cable could be loose.
If in doubt of a solid connection, unplug & re-connect carefully but firmly.
I'm pretty sure your system is not restarting because of the CPU but rather a loose connection or dusty / dirty memory, poor connection & bad or too weak of a battery. Normally I would suggest clearing the cmos/bios jumper on the board but Compaq does not provide the motherboard layout in the user guide. The battery on the board. Is about the size of a quarter. The average battery life is five years. This Compaq was built Aug-2003. One can remove it & check its voltage with a volt meter. It should hold 3volts DC. You can buy a replacement online or ask a friend that owns a volt meter to check the battery. Here is a picture of the battery:
When finished, close the case & plug the power cord in.
That second fan in the system is not connected to the board & that is why there is no speed reading coming up with the software you installed. (This is ok). (Use a small clean paint brush to wipe the fans blades from dust build up, (both sides).
I'll provide Compaq's link for your Presario S5000NX DM171A. (Drivers & software):
Never know when you might need a driver.
Need to buy a battery:
Well this should fix the re-start issue. Take your time.
Post if the problem persist........
Good luck Donna!