I am not an expert, but my understanding is that the ratio of invertase to fondant should range between .1-.3%. The larger amount will make the process speed up a bit. So for a pound of fondant (454 grams), you would want to use .5 to 1.5 grams of invertase. I'm guessing this is about 1/4 tsp, but you should of course do things like this by weight not volume.
As for keeping it cold, I don't think so. You want to make sure you don't add the invertase into a fondant mixture that is too hot (nothing over 150F), but once the final pieces have been dipped storing them at a cold temperature actually slows down the process. So if you want to move the process along keep them at the same general room temp you keep your other confections. Others here with more experience may have more (or better) info however.
Remember, invertase is an enzyme. Enzymes can go bad (if old, or if hot). Enzymes will work much more slowly when they're in a cold environment (ie they will work slower in a fridge than they do at room temperature). Enzymes require some moisture to work (if you've got a formulation without any water in it, for example...), and invertase will only do much for you if you've got sucrose present (ie if you're formulation is with other sugars or sugar free, don't expect much)...
Keeping the invertase in the fridge will extend it's shelf life; however if you've got 10 year old invertase or invertase that spent the first 2 weeks of it's life in an egyptian garage at 140F, it's not going to much matter....
Invertase as an enzyme will continue to work until forced to stop. One way is heat as mentioned above, another is pH, but not a problem with fondant unless you've added acid for a fruit flavor. The final way it's stopped is if there is not enough moisture in the fondant.
Invertase will work until the moisture level of the syrup phase falls below 20%. Most fondants are about 50:50 crystal to syrup (but can range to 60:40, and about 12% total moisture. So if 50% syrup phase, it's about 24% moisture. So you should see significant softening. If less moisture, just a couple percent, will result in minimal or no noticable effect. However, it all depends on all of the above.... age of the invertase, temperature seen, pH, and then moisture level of the fondant and resulting syrup phase. The level of invertase will effect speed to the end result, not how far. A lot more involved but the above should hopefully help.