Sorry for taking my time to get here!
Well I've had 4 rescue greyhounds now, one (Breagha) straight from her trainer, my Dad chose her completely on a whim and took a big risk as we had other dogs, a kitten and a youngish child at home at the time. She was a nervous wreck when she came home, every little thing scared her, she really had known nothing but the race track and her kennel. However with a bit of patience, she turned into a wonderful dog.
The others, Todd, Maddy and Lucy, since her have been easier with regards to settling, as they had all come from a rescue centre. Todd had lived in a home before ours, and he killed their kitten. It was entirely the owners fault, they just went out one day, bought a kitten, and let it loose in the house. Todd was a successful racer...unfortunately, the kitten was killed within a minute. Poor Todd got the blame of course, and as it was friends of ours, they asked if we wanted him. With training, he was completely cat safe within 6 weeks and has been ever since. He sleeps with the cats, lets them share his food, and will happily go on walks with them.
Maddy was a failed racer, she had no instinct AT ALL, she walked in the house, smelt the cats, then went to sleep. She just saw them as another dog I think. Never ever even looked at them in a way I didn't like.
Lucy had never seen a cat before, but she was also deemed a non-racer. However it was different with Maddy, in that she failed because she wasn't fast enough, not because she didn't have the instinct. However I was told her brother was living with cats and was fine, so I took a chance. Turns out she was also fine, she will try to play with them, but she doesn't want to hurt them.
As for rescues, they differ. My rescue checks them with different dog breeds, they have a giant breed, other greyhounds, 3 medium breeds, and 2 small breeds, to check if they are suitable with all. They also cat check them, and child check. They will also do livestock check, and small animals (the woman owns pet rabbits), if you require this.
As for scary things, once Breagha got over her fears, she was pretty solid. Todd is a wimp, it doesn't take much for him to get jelly legs, mostly thunder, jets and big bangs. Maddy was scared of hammers, nothing else, but it was suspected she was beaten with a hammer before she came to us. Lucy is scared of nothing, she has no fears at all. She didn't have to get used to anything, nothing phases her.
The only thing to watch out for with ex-racers is joint injuries. Out of my four, only Todd has had a problem with an old racing injury. He gets a sore shoulder. It doesn't pose a major issue for him, just gets a bit stiffer in the winter and goes away with a bit of glucosamine or Metacam.