I'm not so sure about minimizing any weight using fabric to pack with - it does get heavy - but the whole load will take up less volume if one uses it for "packing material"
As far as wrinkling goes - I always iron/press my fabric immediately before cutting it anyway - and if placed/wrapped smoothly around the items, it shouldn't rumple too badly.
I've never moved so I don't know the answer - are the moving charges by weight and distance - or by weight, volume, and distance?