A couple of points in response to Justone..
Firstly, I'm not disputing at all that you had problems with your grey's toilet training, but in general most greys pick up the idea very quickly indeed, provided they are given plenty of encouragement and clear signals. Ours picked it up within a weekend and this would be normal. If a dog was struggling with continence in the house, I'd want to be visiting the vet, and possibly looking at diet. One of ours was rehomed to the charity after reports of pooing in the house in his previous home, and it turned out the poor old boy had a very sensitive tum, which was being exacerbated by the diet he was on. When this was sorted out, poos were fine again, and all outside.
Am v impressed and of Justone's intelligent hound. All the ones I meet are to put it kindly, behind the door when they were handing out the brains. Except for a very few specialist areas - sofa colonisation, food stealing and running very fast.
The episode with the stuffie is normal grey behaviour - zoomies. Five minutes of mayhem as they run around at high speed - ours does an impressive "Wall of Death" routine between the various beds, usually disembowels a stuffie then collapses in heap for the rest of the evening, and can be found upside down on a sofa, snoring and farting contentedly.
Like Justone, I'd be concerned about the rabbit, but this is where it's important that the rescue is kept informed and works with you to find a grey with a low prey drive that can be small furry friendly. This isn't related necessarily to gender - our bitch has a high prey drive but one of our males is the soppiest imaginable and would probably be fine. Much more to do with individual temperament and background. As a rule of thumb, around 20% of greys can be cat trained and would probably be amenable to fluffy-training.
With regard to being left alone, it's worth remembering that greys in their working lives have spent their entire life with other greyhounds, including being kennelled with other dogs on a permanent basis. Life on the outside is therefore full of new experiences - houses, washing machines, dogs that are not pointy and most scary of all, not being with others 24/7. Usually a rescue will, where possible, bring new dogs into foster care so they can learn about things like stairs, vacuum cleaners and what it's like to have a whole new lifestyle in retirement. Very often, a dog that is prone to SA will be picked up at this stage. However, many greys do successfully make the transition to solo dog and learn about how to be a member of a family. A few don't and will always be more comfortable in a multi dog household (we have one like this). Again, a good rescue will have assessed the dog's temperament and will not place this type of dog in a single dog household. It's also worth saying that greyhounds are thoroughly addictive, and most people I know who start with one, often end up with a second.