I think you'll find it hard to get a resue greyhound , because very few will be unable to resist chading your cats or guinea pig. With possibly terrible consequences! They are usually ex racers and therefore usually not great with small furries. I say usually, because its not always the case, but a cat friendly rescue grey isn't the norm.
I had this problem too. I had two cats but wanted a greyhound. I talked to lots of rescues, and ended up with a saluki x grey lurcher puppy. Because she was a puppy, she was taught to leave our cats alone (I suspect she sees them as special dogs, because other cats she will chase). Most hounds are very very lazy, but will happily walk with you for miles. My lurcher needed long walks when she was younger, but just gets about two 45 minute walks a day now, she could walk all day though.
Off lead, it depends on the dog, how much recall training you do. Some are off all the time, some never. I personally don't think it matters, but a good sized garden will help them be able to stretch their legs, and sun bathe.
Hounds don't have a great reputation for training, but I've found mine fairly biddable. They aren't particularly naughty, and they are certainly clever, just independent in thought. I think you have to persuade them to do what you want!
Illnesses- teeth and dental problems. I brush teeth several a month and have had no problems so far.
Honestly, I love my hounds (my other is an afghan). Quiet, chilled beautiful! What more could you want?
Unfortunately, lots of rescues have lurcher pups come in I suspect that's the way to go for you.