Is the proper storage of tires mounted on rims right on top of each other(flat) or some other way? Should the tire be deflated, does it matter?

 

Keeping them standing up helps maintain the integrity of the rubber. Stacking them on top of one another is not recommended. The bottom tire will take too much weight on the side walls, not to mention the effect of condensation/moisture build up between the bottom tire and the concrete garage floor.

A simple 2x4" frame mounted high on your garage wall works well and keeps the tires out of the way.

Asif

 

I stack mine up, with cardboard on the floor, and also between each tire. No deflation. Never had deterioration or any other problem despite always having extra set of winter tires for all my cars. I put them in my dehumidified basement.