How does an encoder-decoder network work?
It probably won't. The whole point of the training was to encode cat images and thus the network has tried to learn what information is the most necessary to keep to ensure a low reconstruction error (i.e. what separates one cat from another) and what information can it throw away (i.e. what characteristics appear in all cat images and can be discarded).

That being said, a dog image would produce a fairly decent reconstruction because most features are shared between both animals. If you try, however, to reconstruct something completely different (e.g. a car) then it would probably fail.