I think you put too much stock in our societal norms when comparing Twilight to a librarian. She isn't a librarian in the same way you are. The library is more than just a place where people can come in and read or ask for books, but in this case, the library is also her home. It's not even clear if the library is open to the general public, although I'm sure Twilight wouldn't refuse anypony entry as she enjoys sharing her books.
All of the examples you put here are also Twilight with her friends. While Twilight does have many personal flaws that would make it difficult for her to be a perfect librarian (handing rarity a book because she expects everypony to be able to search them as well as she can for example), you can't directly compare her actions with the actions of a normal librarian, because the situations and cultural protocols are completely different. In Cutie Pox, Twilight is not being a librarian, she is trying to help a friend. Twilight has her personal books with her other books because there are quite possibly no free for loan books in the library, it could all be up to Twilight. There is also no confidentiality of a reference interview anywhere in Equestria.

Basically, you're comparing two situations that don't make sense together. Sure, Twilight owns a library, but she's not a librarian, and by saying she's making mistakes a librarian would make is like saying a librarian shouldn't give out a book because people need to pay for them at a bookstore. All 3 (Twilight, bookstore, and library) are similar places, but they aren't the same.