If the game does go in this direction, then dwarves should favor eating old food before it goes bad over eating new food. Otherwise it's a logistical nightmare. Oh, nobles could insist on only the freshest kitten roasts, but aside from them, everyone should do their best to ensure that food doesn't go to waste.
As for storage, global temperature gets pretty flat once you get a few dozen feet into the earth (I doubt that our forts get deep enough for the mantle to start warming things up). Of course, magma would screw that up a bit, as would water, which can "carry" temperature in from elsewher. Water also increases humidity, making mold a lot more likely. Better keep your waterfalls away from the food stores.
Flour, seeds, sugars, and the like should keep pretty much indefinitely. Most foods can also be dried to improve their longevity; drying would involve slicing the food into thin strips and then leaving it in an oven on low heat for extended periods of time. The resulting food would be edible, but would leave the eater relatively dehydrated. This would be a kitchen task, naturally.
Requiring salt to preserve food is a bit dodgy gameplay-wise since rock salt is pretty hard to get unless you have it in spades. I'd favor having some kind of plant, probably surface-only, that would be processable to gain salt. Then you could at least import the seeds from the human caravan and start your own salt-generation industry.
Another way to preserve things is to saturate them with sugar; candying removes moisture much like drying does, and actually renders the food more or less impervious to decay, so long as you don't get it wet. This would use up a unit of sugar/syrup, but would actually make the food more valuable. Typically you'd make candied fruit; in DF's case that'd basically mean the berries and plump helmets, which are the only plant foods that can be eaten raw.