I hope that if you are coming back to post, you truly are reading.
Kids and puppies are a tough mix. I don't blame you for being upset that your daughter was scratched in the face, but Sara's behavior needs to change. You said you are going to start supervising her when she is with the puppy. That is a great start!
Perhaps you can consider making it a project of sorts. Get Sara involved in all the care: have her go with you when you take him outside for potty breaks, scoop the yard, do his food and water, wash toys and blankets, go to the vet, etc. As she takes more ownership of the dog, some of the more inappropriate behaviors on her part may become less. She will see what goes into caring for Shamrock and will realize that Shamrock is a living and breathing animal.
Explicity demonstrate for her exactly how Shamrock should be handled. Demonstrate for her the proper way to pet him, to pick him up if he needs it, etc. Maybe bring one of her stuffed animals and ask her to point out the differences to you between a stuffed animal and Shamrock.
Another thing you can do is ask your local vet or animal store for a good kid friendly book about puppy care and appropriate behavior around a puppy. Make sure that if she does get too rough with Shamrock there are swift and firm consequences. Maybe turn it back on her with language like, "Would you like it somebody pulled on your leg?" (or whatever the behavior is.) Make sure she knows there are consequences for her action and help her to empathize with Shamrock. We call this theory of mind. Little kids don't mean to be cruel, but many just don't understand that what they think and feel is not the same as what others think and feel. If you can get her to see things from Shamrock's perspective, perhaps her behaviors will decrease.
As Sara's behaviors decrease, so will Shamrock's. He will have no need to scratch her because this is a defensive behavior. He needs to learn quickly that she is not somebody to be afraid of or you will have problems for a long time to come.
As for the behavior with the other dogs, that is normal dog behavior. My Matthias loves to pounce on Erie. I will look over and find Erie's tail or ear in Matthias's mouth and he doesn't seem to mind it at all. The only worrisome thing in your other post is the food aggression as this can translate to people and that is something to stop immediately. Talk to your vet or a trainer about safe ways to train it out of him (it can be done.) Good luck!