Ooooh exciting! The most excited I've ever been as adult (including before our wedding!) was the week before we were going down to the greyhound sanctuary to meet and collect a rescue greyhound.
I would agree with greys being the best dogs ever. We've had ours for 9 months now and she is superb. Adapted really easily to life in a house.
Eyes-open sleeping - yes she does, and it's very unnerving particularly when she does it on her back with her head lolling off the sofa. She looks dead. She doesn't have sleep aggression (where they snap when disturbed) but I gather some of them do and I was very careful to warn the kids never to approach when she wasn't very definitely awake.
Sensitive stomach - meh - probably a bit but didn't show any signs of distress when she stole two - yes two - Victoria Sponges off the side (waiting to be taken to the church fete) or half a pound of butter. She's certainly better on one food than another and was very dicey after a lamb bone the other day but it's not catastrophic.
Definitely be aware of the counter surfing tendencies - they are very tall dogs - perfectly tall enough to swipe things of counters and tables.
I would recommend getting a crate for at least the first few weeks/months. It's good to have somewhere for her to go when she doesn't want to interact with everyone. It's an absolute rule in our house that she's never touched or disturbed when in her crate. It's also really useful to have somewhere to put her when you need to have the front door open for a bit. She's fine now but was dicey with leaving the house when we first had her which was slightly hair raising.
Also worth getting a raised feeding bowl as their legs are so long it's more comfortable for them to eat with a raised bowl.
Off-lead - we can with ours if she has a ball to occupy her - otherwise we muzzle. Recall is decidedly sketchy but that's our fault for not working on it enough. You'll probably be fine without a muzzle if she's not raced but I wouldn't risk it until you know her very well.
But yes - definitely best dog in the world. She doesn't bark, she stays in the house happily, will walk as little or much as you like, gentle, calm round the house. We love her.
Come back with pictures!