Does anyone have tips on how to "store" bedding, etc?
Not sure exactly what you're asking.
Don't store it in plastic, it's bad for the fabric, especially cotton. Store in something breathable like cloth (I use pillow cases for my quilts).
I would just do like 14+ suggested and cover the good bedding with something else, like an old blanket, during the day.
As far as the cat tearing stuff up, if you keep the claws trimmed it shouldn't be an issue. You can file the end slightly with an emery board to dull it. You could also try soft claws, which are plastic covers that go over the ends. I have 11 cats here and none of my stuff gets ripped so I think that's honestly a bit odd unless your fabric is already in bad condition. If the fabric has been stored incorrectly it can weaken it and that may be part of your problem.
That's for Linen, but the storage rules are pretty much the same for cotton.