Do these exist? Going past the animal shelter I saw a lot of black cats with white paws. Why not the other way around?
The reason for this is that white is not really a color for cats - it is an absence of pigment. In the cat embryo, the pigmented cells are distributed along the neural tube (the backbone-to-be, so to speak). This pigment then cascades down the sides of the body, much like pouring paint on the cat’s back. The parts that do not get covered by the pigmented cells remain white.
This is why so many cat’s have “tuxedo” markings.
A white cat with black feet is really a black cat with a really big white spot covering most of its body. There are actually three different ways a cat can be white, but only one of them can yield a cat that is partially white.
The first is albinism, which is a complete absence of all pigmentation. Albinos have completely white fur, white skin and pink or pale blue eyes, and often have other problems related to the albino gene. Albinos are pretty rare as the gene is recessive and decreases survivability.
The second is the “self white”. Self white cats are white everywhere and have blue eyes. Self white is dominant, and when present overwhelms all other genetics for color.
The third way to get a white cat is what is called “complete white spotting”. This is a cat which has an underlying fur color and pattern, but which has been completely obscured by white spots. The genes which control the presence of white spots on a color cat are complex and occasionally result in white spots covering the entire cat, completely obscuring whatever color the cat would be otherwise. Complete white spot cats can have green, gold-green, gold, or brown eyes. Sometimes a self white will have one blue eye and one other-color eye (called odd-eyed), or two blue eyes (this occurs when the spots cover the eyes). Spotted white cats with blue eyes are often deaf and odd-eyed whites are often deaf on the same side as the blue eye.
If your white cat has pink or pale blue eyes, it’s an albino. If it has blue eyes and is not deaf, it’s probably a self white. If it has any other eye color, it’s a completely spotted colored cat.
There’s a cat down the street from us who is completely white except for the tip of her tail, which is ginger. I see no reason why a cat could have a white body and black (or blue, chocolate, lilac, cinnamon, fawn, red, or cream) paws.
Spotting should not be confused, by the way, with either Siamese/Burmese/Tonkinese marking, which are variations on the albinism gene (which is why all Siamese cats have blue eyes) and involve a temperature-sensitive deactivation of the pigment gene (warmer parts of the cat end up white), or Van marking, which is not yet understood genetically as far as I know.
For more on cat genetics, see http://www.fanciers.com/other-faqs/color-genetics.html or http://members.tripod.com/~siamkatze/cca/genetics.html.