That's right, they can all have eggs. The fish, the cats, the plants, the rocks, even the genderless clay statues.

[Wait, he probably shouldn't introduce her to the brain-breaking aspects of this world so quickly, should he?]

Plenty. I've got a living ice cream cone that can throw blasts of ice, a lion that can shoot lightning... What they can do depends on their type. There are eighteen of those, and all Pokémon have either one or two.