stacks of kites (during travel w/Friends!) are placed behind the sofa in assembled bags, ready to use. We just pull the sofa out away from the wall sufficient that everything like that fits there.
Eventually you'll have this kiting crap all over deep and around your residence, ... just sayin'!
Either way, remove the tension for storage. Elastics and fabric both stretch out. On most kite designs it isn't too hard to swap out old elastics unless they're sewn along an edge, but once the fabric stretches the tension is gone forever.
Laid open on a floor seems risky for punctures, tripping hazards, and other damage. I'd prefer something hung on a wall over being on the floor just for the safety issues.
You just missed the Lincoln City kite fest last weekend. It was a great time and I got to fly team quads with some fellow kitelifers. I have a kymerra with an ultra lite frame in it, if you want to try it. I think thier still 50% off at Into the Wind? Or if you really want to start spending money there's always quads. I could meet you at Pine Nursery tomorrow afternoon, I have a kymera and a 4d, for duals and a full bag of quads. I don't hnow anyone else in town that flys regularly. The fairgrounds in Redmond has cleaner air and way less people. It's worth the drive.