I believe this is because 'are you seeing her' is likely to be understood to mean you have an appointment to see her, and this is clearly not the meaning of the Russian sentence.
Because pronoun objects are usually placed before the verb. But this is true only for pronouns, nouns usually follow the verb.
Yes, «Вы ви́дите Ве́ру?» and «Вы ви́дите де́вочку?» is the most natural word order.
Note that Ве́ра is declined in the same way as де́вочка, so it takes the ending -у in the accusative case.
I believe "are you seeing her" is not a question about the sight, it's a question about meeting the person, right? If so, it can't be translated «Вы её ви́дите?».
- она = she,
- её = her.
It's more complicated than that because Russian has more forms than English, but I hope you get the idea.