If you half the insulation, it won't half the insulation rating, it will be more like 30%.

But thats if the door already has some nonzero insulation rating. In in-wall insulation just a 4mm gap is about 10% loss in insulation rating.

Uneven partial insulation will result in uneven heat transfer. Uneven zero insulation will make the rest of insulation basically pointless. If your window is open wide then it doesn't matter how good your wall insulation is. Diminishing returns vs exponential loss.

As far as the weight distribution, yes the weight needs to be evenly distributed. If not, then the door can shift in the rails and start binding. Worst case it gets stuck and something breaks.