Wrong. Any animal can be mummified, artificially or naturally--I've seen mummified cats in Egypt, and scientists have found at least one mummified mammoth. In mummies, the original tissue is still present.
Generally, fossilization occurs when the original tissue has been replaced by minerals--turned to rock, basically. It is possible for an animal to be naturally mummified and then fossilized, but it's not necessary.
What doesn't exist is a fossilized cat (or human), or a mummified (but not fossilized) dinosaur.
Not sure what you mean. A simple Google search reveals all kinds of fossilized cats, including the popular Sabre Toothed Tiger, Proailurus, Hoplophoneus, Panthera Blytheae, etc.
Incidentally, not all definitions of fossil include the idea on mineral migration into bone. eg Merriam-Webster: "a remnant, impression, or trace of an organism of past geologic ages that has been preserved in the earth's crust".
Dictionary.com defines it as "any remains, impression, or trace of a living thing of a former geologic age, as a skeleton, footprint, etc. "
Semantics, I know.
Ever hear of the cataclysm of Pompeii?