My Hitachi EC12 does not start at all. Plug it in, turned it on and no electrical humming, no start up, nothing. Pressed the reset. Nothing. Is there a fuse that needs replacing? What kinds of things should I be checking?

if you replaced capacitor exactly like it was remove and no wires switched you have a bad motor.

Check the circuit breaker reset located under the shroud above the dipstick.
If the unit hums but won't turn, first drain the air completely and then try it (If it starts, your problem is most likely the check valve located on the tank)
2) Unplug it and carefully use a thin screwdriver to turn the fan blade - if it spins and is not bound then try it again.
If it still doesn't start I would look at the start capacitor located under the shroud

SAE 5W50 and if available use full synthetic.

This link will take you to the EC12 owners manual if you would like.

http://www.hitachipowertools.com/product/files/pneumatics/compressors/EC12__OM.pdf

I have the same compressor with the same problem. I have temporarily eliminated the thermal reset with a piece of 12 gauge hard wire and it runs just fine. However, I do want to replace it before the warm weather come to the north. I tested the draw at 15.5 amps and the reset was a 17 amp. I am told by someone with superior electrical background that I need to install a 20 amp thermal reset in it's place. I have found that mine runs best when plugged directly into a 20 amp circuit and when using an extention cord I have a 10 guage cord that will carry the load needed to run the EC12. I have also changed to 100% synthetic oil.

Sounds like the pump is siezed or the starter capacitor has burnt out. I would try to get to the armature on the motor and see if you can turn it by hand.
If you can't turn it, remove the head of the pump to see if the valves are ok and that nothing is jamming the piston from moving ( rust, broken rings, bits of valve ).
If the piston is ok, it may be a siezed bearing.
Happy hunting

Sure the outlet you plugged into is working, do not use an extension cord.

If there is power, then there is a black box, where the cord enters the unit., in here is the pressure switch, it tells the unit to work,
unplug the unit open it and see if the plate with the spring moves freely
and then try it again, if not you probably need a new one, I would contact
the company and see if there are problems with the unit and see what they will do for repair, or they might have a better option for troubleshooting this unit.

Single phase motors use a start capacitor and a start switch (that is internal to the motor) for start up of the motor, if either one is not working it will cause this problem, I would advise you take it to a motor repair shop to have it checked out.

try an outlet closer to the electric panel I have had the outlet wire size to small and far from panel and unit only hums let me know if this helps

There should be a pressure switch adjustment under the cover of the switch remove the cover and reset.

The burst of air you hear as the compressor shuts off is the cylinder unloading so it will not have to restart against a high pressure load. If the unloader is electrical it would explain the hum that you hear. This could be a normal function.