It's better without the case. It gets hotter inside the case. You can set motherboard on anything that doesn't conduct electricity. In most cases on an anti static bag. I have it like that, my second motherboard is set on top of a laptop cooler.
Anti-Static bags are slightly conductive. Never place running motherboard on one of those.
It depends on the bag. The best way to check is to use a multimeter before putting it on the specific surface.