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If I cast Restrain from the runes of battle discipline, causes target to halve their move distance until next psychic phase, and I cast it on an enemy flyer with a minimum required move distance (20"), does the flyer automatically crash? Barring of course it doesn't have the option for hover mode.

Most flyers don't have a maximum move of 20" as well. So if the flyer had the option to move 40-60" then it could still move, even at half, its required 20" move.

It can probably just move to it's max move and still meet the min required movement even with the penalty.

you half the movement charistic, which in this case consists of 2 values, so you half them both

Got it.