No, rabbits do not play dead. When rabbits are sick, they show little to no signs of illness. They do not want to appear more vulnerable and weak to a predator. They are already prey and showing signs of illness would make them even a better target for predators.
I'm surprised it even went into your home if it's from the wild!
Rabbits can be scared to death, literally. They go into shock very easily and can get very stressed. The rabbit is a prey animal. No matter how domesticated a rabbit can be or in the wild, rabbits are prey. Those instincts show through. Your cat could have scared him to death. (this isn't your cat's fault, don't worry. it's just their natural instincts. the cat is predator, the rabbit is prey. thats how they both see it in their eyes, because that is just their instincts)
I'm afraid it may be too late for the poor bunny. Don't feel guilty, there really isn't much you could have done at this point. If you aren't very experienced with rabbits, I know this could easily confuse anyone who isn't. But try to take the bunny to a local vet ASAP, just to see if there is ANYTHING to do to help him.
If he survives, I recommend you (or if the vet keeps him, ask the vet to do this) Take him to a local shelter or rescue group. Knowing that the bunny went into your house and you are a complete stranger to it, that is rare. So I'm guessing he was domesticated, someone's pet. And possibly it got loose or lost. Maybe they could find his original home if he's lost. And if he is restored back to health and no home to return to, consider adopting him yourself. But learn about bunnies first. There are many steps to care for a bunny, but if you are knowledgeable about them, you can live a good life together. Rabbits are fun (not to mention just laugh out loud funny) creatures. I love 'em!!
hope i helped!
(i referred to the bunny as a he, because i dont use the word it because bunnies are living creatures)
experienced rabbit person
ADOPT A HOMELESS RABBIT! THEY NEED HOMES AND ARE GREAT COMPANIONS! VISIT THE HOUSE RABBIT SOCIETY'S WEBSITE TO FIND A LOCAL HOUSE RABBIT SOCIETY CHAPTER IN YOUR AREA!
THE HOUSE RABBIT SOCIETY'S WEBSITE IS: www.rabbit.org