Caring for Kneaded Erasers
Since a kneaded eraser works by absorbing the pigment you are erasing along with a little friction to lift the stuff, to keep it healthy keep it away from things it can absorb. Like sand or dust. You also want to keep it dry.
Keeping it off the carpet may not be not so much about keeping it healthy as keeping it from getting stuck in the fibers.
Temperature wise you really do not have to worry. Normal room temperature is fine, it will get soft if you get it really hot, but keep it away from flame or hot surfaces seems obvious. If it gets cold a few minutes in your hands will bring it up to working temperature. I do not think they freeze.
Again, they work by absorbing pigment, though the rubber can get stiff with time (many months as you have noticed). When they get stiff often just stretching them for a while, like stretching taffy, will refresh the eraser. At a certain point they will have absorbed too much and will become useless. It is more a usage thing than a time thing. So expect an unused eraser to last over a year, longer in its wrapper. Keep in mind though, they are designed to be expendable, not archival.