Remove PC PSU From Cage
If you have to ask, then no. It is not safe. The input capacitor is at extremely high voltage (upto 350V for 240V mains, not counting transients). And as you suspect, there will also be electrical noise.

The metal chassis is supposed to be earthed. Besides safety, it also acts to suppress EMI (via Y capacitors), which may interfere with whatever it is powering.