If the GPU is ok you might wanna try without any ram, should give you a different error code. Might give you pointers in the right direction.
PSU testing is typically a bit of a mixed bag. If its really broken you would probbaly notice without any tests. It can get a bit fiddly as one needs to run it under load and you would need to know what voltages are typical for your PSU. Measuring 12.5V for example on a 12V rail (not under load) is quite typical for some PSUs!
Failures are more difficult to spot even with the right equipment. Checking voltages typically only is worth if someone tempered with the cables, you smelled some famous blue smoke or a cable looks suspecious (burn marks).
Most of the time exchanging it for a spare PSU is a sufficient option. You said you found lots of dust, did you try blowing out the PSU? If the dust layer was old enough it can get “conductive” enough to cause problems.
BTW. thats a rather old motherboard, did you check the BIOS battery? You might need a new one!