My oldest desktop is a Dell Dimension 4600. Slow but a good work horse.
Regarding the humming; the 60 Hz is the clue. You may be hearing some resonance, and this may be transferring to the case sides which are sheet metal. The 'DELL' logo is embossed in order to provide a stiffening effect.
Try attaching a piece of flat plastic to the inside of the case cover. This will alter the resonant frequency and prevent the case acting as a megaphone for the transformer hum inside the PSU. (Lead self-adhesive automotive wheel balancing weights will work great).
(All electrical machinery using lams will hum).