I don't think you will find a shop manual for a pro-sumer air compressor. I think something like the parts manual (below)
Repair parts for Husky C602H air compressors
mastertoolrepair.com
and the people on this forum are likely the best thing you will find.
Also, keep in mind that a basic compressor like that one isn't often that hard to trouble shoot once you understand what each part does.
The short version is motor turns on and moves the pump. Air gets pulled in through the air filter and inlet valves. When the piston moves up the air gets compressed and pushed out the outlet valves into the pipe that goes from the head to the check valve on the tank. That's the one way valve that keeps air in the tank from going back to the pump.
When the tank pressure is low the pressure switch turns the motor on, when it's high enough it turns the motor off. When the motor turns off the switch also opens the unloader valve. That's the pisss of air you hear when the pump stops. It's a small hose from the check valve to a valve on the side of the pressure switch.
There are a number of common compressor failures. First, you could have a motor issue (bad capacitors or an over heated motor). You also might have a pressure switch with bad electrical contacts. Those failures are typically going to be failure to start or run.
Typical pump failures are due to valves going bad or a blown head gasket where the intake and exhaust parts are no longer separated. A bad valve might result in slow filling and never getting really filled. The same might be true of a blown head gasket. Sometimes it will result in feeling air coming out the intake.
A bad unloader valve (stuck closed) will make it hard for the compressor to restart shortly after stopping. Basically the motor isn't strong enough to start the pump when pressurized air is in the head. If the valve is stuck open the compressor may fill slowly and you will hear/feel air blowing out past the unloader valve when the pump is on.
A bad check valve will result in the tank loosing air pressure quickly after the motor turns off. You might feel the air coming out of the unloader valve when the pump is off (not the quick pisss after the pump turns off). If the air continues to come out the unloader valve that's not the unloader failing, that's a check valve failing.
Over filling the crank can result in issues and possibly oil being pushed out of the crank.
Finally, some problems are due to a slipping belt. That might be noise and the compressor not able to get to full pressure. Kind of like a weak clutch that gets your car moving but slips on the highway.
Hope that helps!