I stack my extra tires laying flat for my Mirage & Forester. I place a board under the bottom one to keep it off the cement of my garage floor. It's never been an issue in my opinion. My Cooper CS4 tires are about 8+ years old (summer use only the past few years) for my Forester, & it hasn't been a problem.
I'm sure hanging them or running a pipe or pole through the center of the rims to hold them off the ground would be best.
Air-wise, I do nothing when I take them off. Now that I have a TPMS tool, I pump them up & check that the sensors are working before I mount them back on.
The nice thing about having two sets of tires is that I can rotate the tires as I put on the other set each spring & late fall.