Stairs, cats, mirrors, cars, etc. are all things you will have to teach your greyhound. When going up stairs, you will need to place your dogs feet one at a time and help it along. Going down stairs requires a firm grip on the collar and a very slow walk down. It will quickly learn the right behavior towards a smaller animal. Never leave your greyhound alone with your family cat until you are sure they get along.
Greyhounds are kennel trained.
This means that they are taught not to soil their own space. At the track greyhounds are used to the routine of being let out in a turnout pen to answer the call of nature. A greyhound has never had to tell anyone it has to go out. Until you learn each others body language you will need to be patient. It is important to know that greyhounds, although leash trained have to be taught to go to the bathroom while on a lead.