You have summed it up so well. You clearly have your wits and sense of humor fully available to you, as well as the ability to organize your thoughts well.

Let me just ask you a question - is it at all possible that he might have clinical depression and not be "lazy" as in "lazy=bad character trait"? Has he ever behaved differently? Has he ever done the kitty litter frequently enough, and, even without prompting from you? Or, have you always done a hugely disproportionate share of favors to him, without any or almost any reciprocation?

One possible solution to the problem is stopping to cohabit. Then you can use his time selectively for cuddling etc. (see in bold below) and not have to watch him sleep on the couch and be annoyed by that sight.

But you said:

"So why am I with him? What do I get out of this relationship?
- the animals
- cheap rent
- I like hanging out with him, cuddling, etc.
- We do get along really well when things are good."

So you benefit from cheap rent, right? that would mean that stopping to cohabit is not a solution.

Whose animals are they? Are both the cat(s) and the dog his?