Even beyond the whole physical incompatibility issue between cats and rabbits...
it is absolutely genetically impossible for cats and rabbits to procreate (or cats and raccoons, for that matter, for anyone who thought that a Main Coon came from a cat mating with a raccoon. As if a raccoon would sit still for that!) It could not even occur by artificial insemination.
Simply put, and to expound on Mary's explanation:
Each separate species has a distinct set of chromosomes which holds its genetic makeup-cats have 19 pair, rabbits have 22 pair. Each parent donates a chromosome to make each pair. With most species, after fertilization, when the cell starts to divide, there would be an uneven number of chromosomes that needed to 'pair up.' The cell would die. There are some rare exceptions, though. Domesticated horses have 32 pair, and donkeys have 31 pair. Their offspring (mules) have 31.5. Like zedonks and tigons, it appears their chromosomes match up just enough to allow for interbreeding between similar species.
However, the location of a gene is not in the same place on the same chromosome between species. Chromosome A on a rabbit might hold the genetic codes for ear size, hair color and bone configuration, but chromosome A on a cat might hold the genetic codes for vision, tail length and brain size. They could never match up.
As far as Kristina & her "Rampsters": Rats and hamsters are not only different species, but they are a different genus. This means it is impossible to interbreed. That's why Syrian hamsters can't interbreed with Winter White hamsters. However, there are rat-like hamsters-they really do look more like rats, only smaller. It's more plausible that it was a rat-like hamster that mated with a regular hamster.
When it comes to cross-breeding dogs: they are the same species, but different subspecies. They have the same chromosomal makeup. It's the genetic makeup within each chromosome that is different. One parent contributes the chromosome containing the genetic code for "short, curly hair" while the other parent's matching chromosome contains the genetic code for "long, wavy hair." Which genetic feature that actually shows up on the offspring is a whole other chapter. Feel free to google "mendel" and "gene dominance."