I don't know if this would work or not, but for displaying wall-sized quilts you could use two of those 'pincher' type boards per quilt - one at the top and one at the bottom. If the sides of the quilt needed support also, you could use either small appliqué pins or another set of boards. This type of hanging may give you room for a few smaller quilts along the sloped walls.
I also think the rolling/folding tables would be good - or else just set up a large table in the center with the wire or wooden baskets underneath it for storage. The table could be multi-purpose. That would mean a rearrangement of some of the furniture, but it could possibly be done along one sloping wall as well.
Beautiful room :)