No, she doesn't shop. People shop for her (mostly those shops with Royal Warrants....such as Fortnum and Mason (in Picadilly, owned by Canadian Weston Family).
No, when she was born she was still a princess. Her father was only the second son of king George V and therefore she was not expected yet to become queen.
Monarchs are known and counted by their first names, not by family or dynasty. The present Queen of the United Kingdom is Elizabeth II because she is the second person named Elizabeth to ever to be...