Keep in mind that sighthounds are sighthounds and "cat safe" simply means the dog hasn't shown signs of wanting to kill whatever cat it was introduced to in a controlled environment. That doesn't mean they won't want to chase a cat that acts too much like prey or that they one day won't decide that they do have prey drive. I have known Greys who could live with cats in the house with no trouble but if the cats were outside, they would be chased and possibly killed.
Greys are funny, stoic dogs. The ex-racers are used to handling, crating and generally extremely good on leash. Foster homes try to acclimate them to the home environment. Some have issues with separation anxiety when they move to their new homes because they are not used to ever being alone. Startle aggression is not uncommon in them (reacting aggressively when they are startled such as being bumped while sleeping) and has probably been the cause of most failed adoptions I have known of, especially in homes with children. Some can be resource guardy but I'm not sure that is a real common issue. Overall Greys are really good house dogs, generally very calm around the house, not usually destructive and very clean. They are creatures of comfort though and will expect to be allowed to lounge on your furniture
They are accustomed to being exercised regularly and as a breed, they thrive on being able to run. Unlike many breeds, they won't drive you crazy if you don't give them the exercise they want but that doesn't mean they wouldn't be happier with it. Many rescue groups claim that Greys are perfectly fine with daily leashed walks. By that, I guess they mean they won't go crazy or tear up your house if that's all you can offer. But that doesn't mean they are happy with that being their sole exercise. They were bred and conditioned to do one thing and that one thing is run as very fast as they can run. I wouldn't have a sighthound if I couldn't offer them the opprunity to do that on a near very regular basis. Not because they "need" it but because it defines what they are. IME Most ex-racers end up fat and in poor muscle tone.
As for health, IME Greys do best on a raw diet. The breed as a whole is very prone to dental issues and Greys not fed raw all seem to need regular dental cleanings. Since they can be tricky with anesthesia, it's not ideal to have to put them under repeatedly. They are also rather prone to bloat and IME raw feeding seems to really reduced the risk of that. Other issues with them are cardiomyopathy and bone cancer. In track dogs, they seem prone to developing bone cancer at break sights if they've had broken bones. Pannus, an controllable autoimmune related eye disorder has been found in some ex-racers as well.
If you are interested in getting a puppy, you're unlikely to find them with Grey adoption groups and most won't place them in inexperienced homes when they do occasionally get them (they all seem to have this silly idea that Grey puppies are really difficult to raise). There are non-racing breeders of Greys, although their numbers are a lot smaller.