I've never had a greyhound but there are a couple near me and they always look no bother at all, very well behaved infact.
Bitches are wonderful. I have a sweet little cocker spaniel and she is gorgeous ( but so are dogs too )
When a bitch has been spayed they do not come into season.
To help her settle in I would take it nice and slowly, get her a lovely big bed with a nice blanket ( although she will probably go up on the sofa ) reassure her with lots of praise and cuddles, but also leave her whilst you go in a different room for a while so she gets used to it.
Very best of luck - the rehoming people will give you lots of advice and will know best how much your new dog has seen of normal life, and what to do to help her settle. In the meantime, have a look at the Retired Greyhound Trust website for good advice.
Do make sure the dog's been tested with cats, though - some greyhounds are absolutely fine with cats etc., but if they are not it can be disastrous. A normal dog may chase a cat but a greyhound has the speed to catch it
Anyway, well done on getting dh to say yes, and I wish you every happiness with the new arrival