Thank you so much MissBetsy - really really helpful!
My only worry really is the exercise for the dog if lead is always on for walks...
Also, I assume a greyhound can manage a good long walk if the family goes for one at the weekend?
Not too worried about the muzzle - surely you can just say "he/she is an ex racer so will chase and catch small furries..."? I guess I wouldn't like silly cautious parents terrifiedly pulling children out of the way as I approach with a dog with a muzzle though.... but that is only a minor problem (and I am more anoyed on principle by parents assuming a cute little westie, say, with is fine for a child to go and stroke without asking!)
I do worry about other breeds though. In particular, its really important that whatever dog we get gets on well with my parents' very bouncy (but not at all small and fluffy!) heinz 57 (black, size of a skinny lab, lab x collie x lurcher x type of generic bouncy dog!) of a dog. I suppose most good rescues would be willing for the dogs to meet before we decide?
As for the training issues... I think I am happy to accept a nearly perfect dog(!!) My parents' rescue dog for instance CANNOT (and they have tried, a lot) learn to walk to heel properly. Really they have worked for months and, frankly, have given up (they say they don't care... so you pick your battles and this is one they are happy to lose). he is food orientated and have tried doing some kind of clicker training to do with touching their hand and having their hand in the right position he would be to be at heel (I think - I wasn't around for the daily ins and outs) but, when it comes to this, he isn't interested!
As I said, our top priorities are, in order: lovely with children, lovely with us, gets on with my parents' dog, house trained.... and I think we'd be willing to work on anything else.
We still have a few months, so plently of time to do lots of research (and oggle the lovely snoods and collars and everything!)....