If you found between 13.75 and 14.75 volts with the engine running, that only means it is okay to do the rest of the tests. That requires a professional load tester to measure "full-load output current", and "ripple voltage". A common problem is one failed internal diode of the six. That will limit output current to exactly one third of the generator's maximum rated current. 30 amps from the common 90-amp generator is not enough to meet the demands of the entire electrical system under all conditions. The battery will have to make up the difference until it slowly runs down over days or weeks. With a bad diode, battery voltage will still be within 13.75 and 14.75 volts, and it may even a little too high.
GM fuel pumps do typically fail while the engine is running, however, when that causes the engine to stall, it will still crank just fine. You observed you couldn't hear the hum of the fuel pump when you turn on the ignition switch. That's a dandy clue, and it agrees with the rough running engine, but when you include the failure of the starter to crank the engine, that is not fuel pump-related. That points all the clues to the generator. Charge the battery at a slow rate for an hour or two, then see if the starter works.
Most recently you said the starter does crank the engine. Is that after charging the battery? If you still don't hear the fuel pump run for one second when you turn on the ignition switch, that becomes the best suspect.
Saturday, September 2nd, 2017 AT 4:16 AM