Indeed, it simply can't, you've eaten everything in the previous sentence! And now both animals have to starve! :)
Both ways work, Swedish doesn't distinguish. If you're wondering why the main version has does not, I guess all I can say is why not?
Is this specific to one dogs food since it has ('s) or can it refer to multiple dogs food that each dog possess?
Just food belonging to one dog. If many dogs own food, we'd say hundarnas mat :)