Following up my earlier post to answer some of your questions.
Muzzle - I used a muzzle only when first adopted to gauge how they were on a walk. With both of mine, the muzzle was off after a week. If your dog is snapping when out, training is a better approach than continuing with a muzzle. If prey drives worries you, ask the rescue centre how they are - a good rescue will have already assessed this. Neither of mine had any prey drives at all, both were rescued racers.
Collar - I use a collar when out which has my name and phone number on. But I use a chest harness for attaching the lead to. It feels more secure, and I don’t like having the lead pulling on their neck.
Toys - both of mine love small furry toys with a little squeaker or crinkle in. No interest in balls or other toys! All dogs different though. For a rescue greyhound, I would introduce toys slowly, give one, let them enjoy it for a while before getting another.
Pyjamas and coats are needed for the colder months of the year. Greyhounds have short hair and feel the cold. We love Bracken and Paws, and Blas and Co.
An indoor cosy space is a lovely idea, I wouldn’t use a crate. You’ll find though that they greyhound will want to be near you! On the sofa, snuggled up to you.
Separation anxiety - my first greyhound had terrible separation anxiety. No amount of training or behaviour specialists could get past this so I just accepted it. It is what it is. If I went out, I got a babysitter! My second/current is better, I can leave her for a few hours at a time. I walk her, feed her, then she’s happy to snooze on the sofa or bed while I’m out for a while. I have a camera set up in a couple of rooms so I can keep watch.
Length of walks - again, depends on the dog. My first was not interested, would go for 20 minutes max then pull me to a cafe, he loved a cafe snack! Wasn’t bothered about walking really. But when we went to the beach he always got the zoomies. My current loves a walk and I struggle to get her home, she’ll happily go on for a couple of hours.
Final note for now - “rescued, not retired”. And “failed racer” isn’t a thing. Greyhound racing is an incredibly cruel industry. The sooner it’s banned the better.