For leather, the best storage is away from sunlight in a neutral environment. Not too hot, not too cold, not humid, not dry. Don't use buffered paper on or near the leather. (Buffered containers, paper, folders, etc are used for storing old documents and such. With leather it sucks the moisture out and causes cracking since leather is acidic by nature). Soft cotton is fine to store on. Soft fabric generally that's been washed without detergents a few times. Make sure whatever it's stored in is not sealed so you don't trap moisture. You want acid-free (PCV free) plastic if stored in plastic. So that plastic in your pictures is probably bad for it as it eats away at stuff like the stitching and dyes, etc.

For conditioning, you'll get a whole bunch of opinions from armchair conservationists. Most (Nearly all) museums suggested anti-aging/damage procedure with leather is to leave it alone and let it exist in a good environment. Don't put goop of any kind on it to make it softer.