You better post some pictures when you bring home that new doggy!
Just to let you know, most Greyhound rescues will temperament test and let you know how they are with cats, small animals etc.
My brother's dog was suggested to be in an "only dog" environment. But she still plays very well with other dogs. When he first got her she tended to be a little bit snippy and intimidated by other dogs but she's improved a huge deal. She is fine with small animals 90% of the time. When it does become an issue is if the small animal bolts, then that prey drive kicks in. There's never been an issue with her and our indoor cat, but outside my small dog tends to run like a little idiot and has a peculiar little hop to his run.. which makes him the perfect imitator of a little rabbit.
Some definitely aren't ok with small animals/cats, others will be fine, and the rescue volunteers/fosters would be the best place to start asking those questions.
You have to be a bit patient with them in the beginning. Although they tend to be great with grooming (the only dog that has actually sat patiently to let me paint her nails

) they usually aren't very well socialized and are used to their own space for sleeping etc. One of the biggest issues is guarding crates or bedding, or sometimes getting snippy if they're awoken from sleeping. But those are definitely things that can be worked on and improved dramatically.
My brother's dog is literally like night and day since he got her and she's suuuuuuch a lovebug. It always breaks my heart to think she had never really experienced a real bond before, but you can see it in her eyes the way she looks at my brother she just loves him with every fiber of her little being.