Hi! Welcome to Yorkie Talk!
To answer your questions :
If only one of her parents is registered, then she can't be registered. The breeder actually has to register the litter with the AKC and then give each new family paperwork to register their puppy.
If DG's tail wasn't docked as a newborn, she will have a long tail for life. After 3 days of age it's considered an amputation and would be cruel to do to her. I'm sure she looks adorable with her full tail
She's only about 8 weeks old, which is very young. She will just gradually grow her coat. It takes time. Some grow their coat quickly, others are very slow. My Belle has only had one very little trim and her coat is still a few inches from the floor at 15 months old.
I wouldnt worry about taping her ears at such a young age. Just shave the hair on the top third of her ears and they will start to stand up.
Enjoy your little girl.