Please dont worry, little puppies tend to get the grumps a bit when they are sleeply.
Our Rotti puppy growled and snarled at me quite a lot when she was very little, just down to being knackerd. I caught her doing it, gave a gentle shake of the scruffs like her mum would have done and put her to bed.
They grow out of it, my mum's three cavaliers have all done it and they are the most gentle of breeds
I would just warn your little girl that puppy can and will growl and so she understands how to deal with it. Dont be afraid and if you show your little girl how to move out the way etc everything will be fine.
personaly i think the dog would be a lot happier in a child free home. its obviously stressed by kids. no dog is 100% bomb proof RE temperment, but i have kids and like to work at 99% ish. my young children are never left unsupervised with my dogs, if my dogs behaved like this around my kids they would be gone, it too bigger risk to take imo.
My Springer Spaniel behaved in exactly the same way when we got her, however she was only 3 months. She didn't like my daughter one bit, although she was perfectly fine with my son, who is younger. It was definitely an "Alpha" thing - she saw my daughter as a threat and was trying to dominate her. It was discouraging as we had recently lost an old Springer who we loved to bits and was gentle in every way.
What has been said on this thread about getting your daughter to feed the dog (and make it sit and wait for the command to eat) is good, and if you read "The Dog Listener" by Jan Fennell, as someone else said, it should give you some pointers.
In the short term I think the behaviour will continue for a while, I doubt you will cure it in a matter of weeks or even months. In our case, the dog is now 4 and almost never growls at my daughter now, she has largely gotten over it.
I had a collie x greyhound when I lived with my parents. When I left home, the dog stayed with my parents as she was about 12 years old and happy there. (I did take her for a while to my new home, but she didn't really settle). About 2 years later, my nephew went to live with my parents. He was about 5 at the time and the dog hated him. If she was laying in her bed which was in the hall and he went anywhere near her, she would growl at him, and if he didn't back off, a couple of times she snapped at him. Nothing really vicious, but she would have a go. If she was out of her bed and he went anywhere near her, she would just walk away from him. I would say that they were never friends, Judy eventually went deaf and got quite grumpy, but she learnt to live with Dan. She would never let him near her when she was in bed, but would often lie with him if he was laying on the floor watching tv etc. Most of her problem was that she wasn't really used to children. My sister was about 5 when we'd gotten her, but the dog was only 18 months herself so they had grown up together.
My bassett hound puppy did this and ripped my mums arm to shreds one day, I took him out and shot him!! Totally not able to rehome something like that totally our fault as got from puppy farm in Wales who drive up with 2 to choose from hard lesson learned will never do this again!
Echo replies about dominance/pack thing. Always make sure you (and your daughter) go through doorways in front of the dog too. It is a tough call, greyhounds are usually a very docile and biddable breed with few demands on the owner, in your position I would be tempted to send her back and try again. There are thousands of dogs looking for a loving home, and a great may of them are fine with children. We had a rescue lurcher once who was lovely to us, but one day flew out of the back door and sank his teeth into my friend's arm! We had him pts, it is simply not worth taking risks with dangerous dogs. The rescue centre were in full agreement. I would be very watchful with this dog. Good luck with it all!