Marilyn, I've always kept mine in the original bag, which I set inside a plastic trash can with a locking lid. I've never had any problems with them going bad or getting mold on them, and no critters can get in that way.
Not sure if the rabbit pellets will do any damage or not. Hopefully not! Maybe someone else can help with that, since I've only used the horse pellets. Maybe some extra water would help? I don't think that all pellets are created equally for rabbits, so check to see if the ones you have are loaded with salt. Maybe they aren't. Since I don't think rabbits use a salt lick (mine never did), that is probably why salt is added to theirs. Horses like salt licks, and don't need the extra salt added to the pellets.
