Strange noises from the computer

I have a computer with 2 fans , one for the processor and one for the PSU.
Occasionally I get noises and there are 2 kinds : one is a squeaky noise and
the other a grinding noise. The squeaky noise sometimes appears every now and
again and eventually becomes continuous. The grinding noise appears out of
the blue and is continuous right from the start. The 2 kinds of noises never
happen at the same time.

The interesting thing is that when either noise happens and I start some
demanding computation running (one which occupies around 100% of CPU time of
one of the two cores) then after about 20-30 minutes , whichever noise was
happening stops and does not return for as long as the computation is
running. I can only hear a very soft whirring sound then. When the
computation stops , the noises may not appear for some hours or days but
eventually return. Normally the computer does not run anything demanding and
the cores work close to 0%.

What I mentioned is the only patterns I have been able to notice related to
the noises.

There are several questions :

1) What causes the noises ? I can't think of any other source apart from the
fan(s) but which is the mechanism which causes these symptoms ? In particular
that the noises go away when one or both cores work hard but otherwise
disappear. Putting the cores to extra work , I assume makes both fans run
faster but which mechanical reason would cause slower rotation to produce
noises which go away with faster rotation ?

2) Is there a way (BIOS or something) to make the fans run faster without
giving extra computation to the cores ?

3) Is there a way to find out which fan causes the noise ? It could be that
one fan does one kind of noise and the other fan the other kind of noise. I
opened the case once when I could hear the grinding noise but I couldn't tell
which of the 2 fans was the cause. Is there a way (like some appropriate
hardware) to make the fans work without powering up the computer ? The PSU is
hermetically sealed within the PSU enclosure but the processor fan is
separate and there should be a way to connect it to something to make it
rotate.