The whole We Were Here series is marvelous. Asymmetrical co-op puzzle games. My friend and I’s recent games list looks very similar to this.
We also do a lot of single player games with one of us streaming over discord. When it’s a slow-burn puzzle or mystery game, it doesn’t really matter who is actually controlling.
For those types, I really recommend Return of the Obra Dinn. We’re currently working our way through the entire Frogwares Sherlock Holmes collection. The old ones are so terrible, which is a greatness all by itself.
Rabbits I’m not sure. Cats naturally want to bury their leavings, so they actually train themselves just fine. Show them where it is, keep it clean, and 99% of the time that’s that.