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Given your description your best bet is probably peel and stick vinyl tile
It's pretty easy to install and won't harbor mold or moisture like a carpet might
Temp changes should have little effect. It won't lift as long as it's set on a clean, sound surface
new 1/4" plywood underlayment is best
What's under the carpet now?
One other option if this is little used and mostly storage would be to just paint the floor.
yes but it's a bit pricey for a shed
Once it's down it acts much like vinyl sheet goods (lino)