Is it okay to turn key without transmission fluid (just to put in neutral)

No. The damage you could do would be from running the engine for more than just a few seconds. The potential damage would be to the front pump seal. It is normally flooded on the back side with transmission fluid to keep it lubricated and sliding freely around the hub on the back of the torque converter. You aren't going to burn that seal instantly as there will always be some residual fluid on it.

Same with the rotors in the front pump. They'll be pushing air into the rest of the passages, but that air will be expelled on its own later. The rotors also have residual fluid to lubricate them for a little while.

None of the other rotating parts will operate without fluid pressure, so there can't be any damage done to the clutch packs and their internal seals.

Yup. You're over-thinking this. There are computer controls running parts of the transmission. That includes solenoid-controlled valves, and pressure sensors and switches. All of that requires fluid pressure that can be measured, can be switched on and off to be directed to different passages, and can apply pressure to the clutch packs to engage them. The only way to get that pressurized fluid is for the engine to be running so it can run the transmission's front pump.

The Transmission Computer knows it can't expect to try to operate any part of the transmission or see signals showing up from pressure sensors until the engine is running. Turning on the ignition switch to "run" or "accessory" is a normal action, whether you're sitting there listening to the radio, or you're preparing to start the engine, but paused in the "run" position for a while. At that point, no computer is even aware there is no fluid in the transmission.

Nope. Fuel injectors, ignition coil and fuel pump are all turned on by the automatic shutdown, (ASD) relay. That relay turns on for one second when you turn on the ignition switch. You might hear the hum of the fuel pump for that one second. Fuel pressure should hold for weeks, but often it bleeds down too low for the engine to start. That one second is to run the pump just enough to insure the fuel pressure is up for starting.

After that initial one second, the ASD relay doesn't get turned on again by the Engine Computer until the engine is rotating, (cranking or running). It knows that by the signal pulses it receives from the crankshaft position sensor and the camshaft position sensor. There's no pulses being generated when the engine isn't cranking, so the ASD relay and everything it powers is off, even though the ignition switch is on.

Hard to say what the noise was that you heard. Drive shafts will be turning when you push the vehicle. Sometimes rear brake shoes drag a little and can cause a brake drum to rumble. You won't hear that while driving.