There might be several ways to handle the protection problem. I'll try to give it some more attention when I have more time. The simplest would be to just put a diode in series with the positive side, but then you'll need the power supply to be slightly higher than the target operating voltage; and if you want to operate something like an A/D or D/A converter off the same supply, you'll probably want it to have its own reference, even if it's only an 8-bit converter. We do have portable products for private pilots, for the ones flying ultralights or vintage or other aircraft which may have no electrical system, and in that case I do put 1N5817 diodes in series with the 9V batteries; but there's a switching regulator after that that kicks the voltage up to a tightly regulated 12V for the audio circuits and another that takes the 12V down to 5V for the microcontrollers. It's a little different for units that are powered by the aircraft power (which may be anywhere from 11V to 30V, plus spikes). The next simplest way to handle the protection problem might be to put a fuse in series with the positive and a diode from there to ground, so in the rare occurrence that you do connect it backwards, the fuse blows.
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Electrons will happily go either way, as long as the gate-source voltage is above the threshold.
It depends on the internal construction. MOSFETs are usually not symmetrical inside, and the dimensions and doping of the source and drain sides are different. Even if it works, the gain may be super low in the reverse direction, or you may run into other problems. I suppose the way to find out for a particular part number is to try it, since the data sheets don't characterize the operation in the reverse direction.