Fish are cold blooded creatures and therefore, their body temperature is the same as their surroundings (unlike warm blooded animals that have a constant body temp.) Technically, fish do not hibernate like, for example, a tortoise would but when the water temperature drops, the fishes metabolism slows down accordingly. Some pond fish such as golden orfe do not mind the cold at all (but they hate the warm) yet koi are the opposite and hate the cold (and love the warm water)
The reason we break the ice....or keep a small area open is to allow gas exchanges (nothing to do with "breathing oxygen" as cold water contains a lot of dissolved oxygen). In autumn, the pond often gets lots of dead leaves from the trees in the pond as well as dying pond vegetation. This sits on the bottom and decomposes causing various harmful gasses to be produced. If these get trapped under the ice, they can poison the fish. Well that's the theory anyway however i doubt the pond stays frozen for long enough for any harm to occur. If you do need to break the ice....use the hot water methods above, if you just crack the ice, the shock waves travel through the water and can scare the fish.
As for feeding, i will feed my koi all the time they want it. Cold fish won't eat, simple as that but sometimes, certain fish such as koi which are generally greedy, will feed in the coldest days. feed small amounts and if they don't want it after 10 minutes, remove the food and try again in a few days. Fish can go several months without food, relying on stored body fat.
The bit about food rotting in the fish's stomach is kind of true...obviously the metabolism has slowed down and so has digestion but it hasn't been proved to harm fish and not many people lose fish the day after feeding them in the winter.
So i hope this helps a bit.