Trying to trouble shoot an issue here. When I plug my tower in (the CPU) I get a relatively noticeable electrical humming noise. I do not notice this on my other computers in the house.

This is while the computer is "off" and just plugged in.

Machine was recently shipped ACROSS country with original foam cushioning and lots of BUBBLE wrap. Did not notice any other damage. PC does BOOT up fine. I'm worried about the noise because the computer is "off" while this is happening (although I understand that the current is still going to the device).

I'm leaning towards a hunch it's a power SUPPLY issue. Any other ideas?

If it is the power supply, can it cause any damage to the computer or data files if it goes out before I can get it fixed?
Power supplies do this when they are unhealthy. Sometimes the electrolytic capacitors when failing will start to make this noise. I'd replace the power supply to avoid a fire, although the powersupplies generally have fail saves in them to prevent fire, but I have seem them fail and STINK up the place.