A few days ago I bought and installed a new power supply (). After turning on the computer and verifying it worked, I leaned over the case to tighten the case screws, inevitably breathing in the exhaust from the power supply. It had a metallic smell. The next morning I had a slight cough. Today, the cough has gotten worse. I don't feel sick, but I have a moderate dry cough. Do you think this could be caused by the power supply? Or is it just a huge coincidence?
I would say it's just a huge coincidence. There might have been dust inside that flared your allergies or something, but I doubt it was the PC itself. I suppose that maybe it it had a blown capacitor and the vapors were streaming out that it would do some harm, but it's highly unlikely. It's that time of the year to be getting sick anyways.
PSUs dont have an "exhaust" like a car does, if it was a brand new unit what you were smelling was a bit of residue left over that just needed to either dry out or evaporate, its not a problem and certainly wont make you sick, after a few days there wont be any smell coming from it as there are no chemicals that get released as it runs, its just air passing by some heatsinks.
Psu's, powerdrills, vacuum cleaners and most electrical motorised appliances in fact, have grease in them,
the smell/taste you experienced is merely excess grease burning off and will dissipate after a short time, dependant on how much use the appiance gets,
its normal and nothing to worry about, and almost certainly nothing to do with your sore throat