If the attacked character is KO'd, do you place the generator in a square adjacent to the attacker?

You can if the squares are adjacent to the target's square. Whether the target is continuing to occupy that square or not doesn't matter.

It must always be adjacent to the square the target was in, regardless of whether or not they end the attack on the map. I believe if the target is knocked back, it is the square the target was hit in and not the square they ended in. Other note: it is adjacent to the square, so it cannot be placed in the target's square, but can be placed in the attacker's square if he is adjacent to the target. Also, keep in mind, squares with ladders connecting two elevations are not adjacent.

Thanks for adding the knock back info