Wildfig, thank you for your kind words. 
One of the reasons we got greys was that DH has asthma which is triggered by cats, horses and long haired dogs, so that even the long haired lurchers I adore or a deerhound (swoon...) would have been quite out of the question. Greys DO shed but are generally suitable for someone with allergies/asthma. However, you will need to step up housecleaning a gear, in line with general anti-allergy advice. So, I'd advise things like daily hoovering, regular damp dusting, minimise the mite farms (cushions, etc) have soft furnishings that you can wash regularly and hot (this also breaks the dog flea cycle so has dual benefit) and most importantly say goodbye to carpets and hello to wood, tiles, laminates, lino or similar that will not trap hair, dust, mites, flea eggs and other assorted oomska. Greys are generally very clean compared to many other longer haired breeds of dog but as with any pet you should be prepared for housekeeping incidents involving mud, blood, vomit, poo and wee in the house at some point. One of ours is also a spectacularly messy eater, and another has the charming habit of leaving little presents for us in odd places (piece of tripe on the pillow was one of her recent standouts).
More generally, most reputable greyhound rescues will be able to match you beautifully with a suitable grey. In general terms, greys tend to be very gentle with people, being used to handled, travel well, are good on the lead and contrary to myth don't need miles of exercise. In fact if you were looking for a very "full on" dog that bounced and bounded and chased sticks and panted and gololloped then a grey is probably not for you. They will colonise your sofa, usually upside down with all four legs in the air, sleep a lot and then have little bursts of fun (we call this "doing zoomies") when they race around grinning like loons. They are complete love sponges who adore being cuddled, but are famous for having toxic wind that can strip paint and should be banned by the Geneva Convention. They are very healthy dogs in general and long lived - usually well into the mid teens. Often need teeth attending to when coming off the track but otherwise OK. The only other downsides are that they are very accomplished thieves (your kitchen counters will rapidly be super tidy) and SOME, not all are cat chasers or not good with other small furries. However, many are, and happily co-exist with cats, rabbits and a variety of other pets. Again, your specific rescue will advise on this. Recall (and obedience in general) is not their strongest point - in fact from a safety point of view remember that they are sighthounds and will fix on/chase somehting of interest from a long distance (that you may not even have seen) and will be oblivious to anything between (roads etc) so off lead play should be confined to safe areas.
They make wonderful pets and companions, are very addictive and you iwll also be helping make a home for a retired athlete who through no fault of their own, would otherwise be facing a needle, a bullet or worse.