Refrigerating will keep them from pupating and maturing into adults. If you just want you tub of worms as food and don't want to breed, then keep 'em cool.

I've only really had success with breeding mealworms (my waxworms make many moths, but no young yet). I wouldn't bother with a sponge for moisture, you can just drop in a slice of carrot, celery, kale, lettuce or whatever veg you have on hand every couple of days. I use a mix of oatmeal, wheat germ and wheat bran whizzed up in a food processor as food/substrate. You can keep a couple of bins, one with worms and one that you move the pupae over into for them to mature into adults and breed. Room temperature is fine, more heat will speed up the breeding process. They are hard to kill even with neglect- I've read more than one story of people who forgot about a shoebox sized colony for a few months only to find it filled with young mealworms.