In Australia, nothing is as it seems. Humans and dingos are far from native. There are giant birds that look like dinosaurs and can gauge your guts out with nasty, long claws. Even the happy-looking, hopping national animal of Australia is perfectly capable of drowning your dog like some sadistic, pouched psychopath.
I keel you.
The most normal animals in Australia are bandicoots (order Peramelemorphia). They are small, furry critters that eat almost anything. Bandicoots are distinguished from another group of small marsupials by their tendencies towards eating grass as opposed to animals, as well as their shorter legs. The word "bandicoot" came from a word that roughly translates as "pig-rat"- an adequate name for a long-nosed, small mammal like that.
"Anteater-rat" would also have worked. This is the aptly-named long-nosed bandicoot (Peremeles nasuta).
Bandicoots exhibit the closest thing that marsupials have to normal births: Instead of being pouched, baby bandicoots stay in the womb, attached via a small psuedo-placenta. This is pretty close to the placenta found in placental mammals, but not close enough to move the bandicoots out of marsupials.
A fair amount of bandicoot species are threatened by (as one might expect) introduced species. Cats have a field day with not-quite-mice. Foxes, adaptable as they are, see no problems with eating any furry things that they can catch. Many bandicoots are now considered endangered.
Despite being the most normal mammals native to Australia, bandicoots are still portrayed as being not quite right in the head. The most popular example is, of course, Crash from the Crash Bandicoot games. Really, does this look like a sane animal to you?
Then again, he's also supposed to be a genetically-engineered freak. That explains that.