you can put Lugols in the fridge but not SSKI . actually, its not necessary to put any of it in the fridge. room temp and dark and air tight. those are the main things. if you put SSKI in the fridge then KI will fall out of solution and crystals will form in the bottom of the bottle. it returns to liquid once it warms back up.
as for shelf life, everything i sell should be good for many years, properly stored. the exception is lugols. it can last for years as well, but not in a rubber dropper bottle.