The power supply fan is very important, especially if you have no other fans in the case (apart from the CPU fan).
If it fails then you will find that your system overheats and randomly crashes. If left unattended for long enough, your PSU could also suffer damage from the heat, and may even catch on fire.
It's just possible that some weird ACPI setting has turned off the fan; you can check this by booting with the command-line option
You could also look in the BIOS settings to make sure that it hasn't been turned off there.
It's more likely that your PSU has developed a fault. Unless you can take it apart and test it, I would suggest that it would be worth replacing it.
The power supply button fault could be caused by any of the following:
- not plugged in to the motherboard (or plugged into the wrong place)
- dust on the contacts (you can fix this with tweezers provided the computer is turned off)
- the button could be disabled in the BIOS or by a jumper on the motherboard.