There’s a common trend among domesticated animals—they tend to develop more variegated coloring than their wild ancestors.
One theory is that domestication affects traits that involve early embryonic development, and that have a lot of associated changes beyond what was actually selected for (i.e. tameness).
Another possibility is that, in the absence of selection, animals naturally produce more variation in coloration than is seen in the wild—but natural selection is continually weeding out those that are disadvantageous in their current environment.
Little mice chasers started coming to human habitations when humans started building silos to hold grain. The humans saw the cats were useful and kept them around.
It might, but exceedingly rarely. Albinism and melanism occurs in many taxa including species in the genus Panthera (which include lions and tigers). These are rare traits in general because they are selected against for one reason or another and don't proliferate in the wild.
Another possibility is that these phenotypes are linked to genes that have other downstream developmental effects that lead to non-viable fetuses.
Another fun one (possibly) is that some of these coat traits are controlled partially by the environment/the animal's spatial relationship with the environment. Siamese cats have the characteristic darker coat colors on their extremities (ears, paws, tails, snout, this is called acromelanism) because the melanism is controlled by lower temperature. If an animal never experiences a low temperature, that phenotype might never be expressed.
These are all possibilities. I'm a biologist but I don't study felids (cats and big cats).
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