The only way I could see your starter doing anything and it not being obvious that it was the starter is if the solenoid is not pushing the pinion out to engage with the flywheel. I guess if you put hot to the starter solenoid post the solenoid would kick the pinion into the flywheel, but I don't know if that'd make a continuous noise (although I'd imagine it would be unhealthy for the solenoid). Other than the starter, the intercooler pump and the fuel pump are the only other things I can think of that would be making any sort of electrical noise in the back of the car. Then again, I have my A2W intercooler and fan in the back too, so if yours is back there, check that, but also I'd think you'd have some air movement to help identify that.
I wouldn't think that swapping any of the wires on the two posts of the starter would cause any issues you're describing. As others have laid out, the hot post on the starter is supposed to be where the battery, alternator, and power supply to the fuse box all come together. I think the solenoid post on the starter is just a solenoid that grounds to the starter body/engine block so putting power to there not through the crank relay would cause the solenoid to kick the pinion into the flywheel but not do anything else.
I can crank my '06 LSJ without the fuel pump connector connected to anything. I'd guess an '05 is the same and I don't see anything in schematics that would indicate otherwise.
I'd start with identifying for sure what's making your noise. Is your A2W heat exchanger in the front with a fan? I used the fog light power wire that DF ran to the front and back of the chassis harness to power my A2W fan. That means when I hit the ACC switch on the DFKC-supplied switch panel, the fan turns on. I did have a small missed solder on one of those switches, not related to the HX fan in my build, but I could see a different missed solder there turning the fans on with the key switched on.
But for sure identify what's making your noise first. Then we can go into if the PCM is satisfied with things to give you a crank. If your instrument cluster isn't lighting up, decent chance it's not satisfied.