Can I replace a faulty capacitor? The one in my unit still hums even after the power is disconnected. IS it still electrically charged if humming without power?

If it is humming then it is probably your contactor not the capasitor, and yes you can change it yourself. Has 2 screws holding it in place and it is as simple as taking the wires off one and putting it on anouther. And for you outher question yes the capasitor will hold power.

When you disconnect the power to your outdoor unit it does not cut off the power to you indoor unit, therefore when the thermostat calls for cooling, the outdoor unit may hum even though the power is off.

And like Joshdta said above it is more likely to be the contactor.