Greyhound mumma here! And definitely agree greyhounds make the best pets!
We've had our boy just over a year now and wouldn't change him for the world. He's such a character and so full of love. I can't comment on the cat front as our boy would scale a 6ft fence to catch a cat if he saw one, but our learnings so far have been...
The scream of death will chill your blood the first time you hear it...honestly nothing can prepare you for it and you will expect to find limbs hanging off...invariably it'll turn out to be a stubbed toe! Silent injuries are by far the worst, we can usually tell he's hurt as he'll just run to us and demand to be cuddled better. Our biggest injuries to date have involved zooming into fences that he hasn't realised are there and eating mouldy food that we thought was out of reach during a fridge clearout. If your bin is in sight I'd also recommend hiding it!
Equally there is no smell like a greyhound's butt! Ours can clear a room in seconds and tends to wait until we have friends over or are out in company before unleashing his special brand of chemical warfare!
No work surface will be safe! Our boy will merrily counter surf given the opportunity, often evidenced by a hotchpotch of items being hidden in his bed when we return. We've basically learnt that nothing is safe from his clutches if he wants it and that he has slightly klepto tendencies but generally he just likes to keep them safe rather than trashing them. He also loves soft toys and will steal them wherever possible...sadly he's not quite as gentle with these and will usually leave a trail of stuffing in his wake so worth bearing in mind if you have little ones about.
Recall can be patchy and prey drive is generally pretty strong (although this can vary hound to hound). We don't let our boy off lead unless we're in an enclosed area as we know his recall isn't 100% and we have no hope of getting him to pay attention if he spots a rabbit or squirrel as he becomes blinkered to anything else. This is getting better with training, but for some it's nigh on impossible to train a recall response over their natural prey drive.
Greyhound cuddles make the world a better place! Our boy is often convinced he's a lap dog despite his size and loves a good snuggle! It's particularly special when he comes over for a lean. Don't be alarmed by the chattering of teeth, it's a sign of contentment. As is nibbling/nipping or "knitting" at arms, sleeves or in our boy's case duvets! Although he tends to be worse if I'm wearing a baggy jumper/batwing style top. Usually it's cute, but sometimes he can go a bit overboard which hurts a little but is usually rectified with a little yelp.
They do feel weather extremes worse than most dogs, so invest in a good wardrobe. We found Redhound to be great for outdoor coats and fleeces. In summer heatstroke can be a worry so we tend to wet a tshirt and pop it on him or soak a tea towel to lay over him whilst he sleeps. We also have a paddling pool for him to play in.
Greyhounds do have a habit of sleeping with their eyes open and can easily startle if approached whilst asleep without warning. Our boy thankfully isn't too bad for this, other than the occasional jump, but I have heard of owners being bitten by a startled greyhound so it's worth being mindful of not doing anything that might surprise her.
If you have any glass patio doors or the like you're best taping them with a criss cross of masking tape for the first few weeks as greyhounds in full zoom don't tend to register glass and can make for a nasty accident. We tend to avoid shutting the door when ours goes out for a wee, much for the same reason.
I'm sure there's a million other things I've forgotten, so will update if anything else comes to mind. Best of luck with your girly!