Thanks for the info, Skuzzy. Am I right to suggest that the fan action is about the full speed fan test not ending because of the lack of the feedback signal?
Puma, there's basically no cost effective way to test the power supply. There's cheap "testers" available but all they do is to measure the starting time and the voltages of various plugs at the same time. Real testing requires oscilloscopes and artificial loads and other techie gear. At some point someone claimed that the price for a thorough tester would be some $ 50 000...
If you have a multimeter, set it to measure 20V DC and check the voltages of as many different cables as you can find while the computer is running. Black is always ground, yellow 12V and red 5V. The other colours vary from 3.3 to 12 volts. If the multimeter prongs are thin enough, you can even measure from the rear side of a plugged connector. Note that if you unplug the PSU from the motherboard it won't run at all unless you connect the green wire to a black one with a paper clip or a piece of wire.
As for safety, don't open the PSU itself as the lethal voltages are inside it.
A PSU from 2009 may not be up to the task anymore. Back then the 12 V line could have been split to several rails so a modern video card may not get as much amperage as needed which may cause the PSU to exhaust to death.
If there's a local builder shop near you, ask if you could lend/hire a spare PSU for testing purposes. Again, most anything with the right plugs will do for a short time boot test. If your system boots normally with another PSU, you'll know what to buy next.